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3.1 methods of division

The document discusses polynomial division algorithms. It introduces long division and synthetic division as methods for dividing polynomials. Long division is analogous to dividing numbers, while synthetic division is simpler but only applies when dividing a polynomial by a monomial. The key points are: - Long division allows dividing any polynomial P(x) by any polynomial D(x) to obtain a quotient Q(x) and remainder R(x) such that P(x) = Q(x)D(x) + R(x) and the degree of R(x) is less than the degree of D(x). - Synthetic division is more efficient than long division when dividing a polynomial by a monomial of the form (

Methods of Division
Methods of Division The questions of factorability and finding roots of real polynomials (polynomials with real coefficients) are the main themes of this chapter.
Methods of Division We start with two division algorithms for polynomials of a single variable x; long division and synthetic division. The questions of factorability and finding roots of real polynomials (polynomials with real coefficients) are the main themes of this chapter.
Methods of Division We start with two division algorithms for polynomials of a single variable x; long division and synthetic division. The questions of factorability and finding roots of real polynomials (polynomials with real coefficients) are the main themes of this chapter. Long division of polynomials is analogous to the long division of numbers and it’s a general division method for P(x) ÷ D(x).
Methods of Division We start with two division algorithms for polynomials of a single variable x; long division and synthetic division. The questions of factorability and finding roots of real polynomials (polynomials with real coefficients) are the main themes of this chapter. Long division of polynomials is analogous to the long division of numbers and it’s a general division method for P(x) ÷ D(x). Synthetic division is a simpler method for dividing polynomials P(x) by monomials of the form (x – c), i.e. P(x) ÷ (x – c).
Methods of Division We start with two division algorithms for polynomials of a single variable x; long division and synthetic division. The questions of factorability and finding roots of real polynomials (polynomials with real coefficients) are the main themes of this chapter. Long division of polynomials is analogous to the long division of numbers and it’s a general division method for P(x) ÷ D(x). Synthetic division is a simpler method for dividing polynomials P(x) by monomials of the form (x – c), i.e. P(x) ÷ (x – c). Synthetic division is particularly useful for checking possible roots or finding remainders of the division.
The Long Division D(x) P(x) Dividend Divisor
The Long Division Set up for the division P(x) ÷ D(x) the same way as for dividing numbers.D(x) P(x) Dividend Divisor
The Long Division Example A. Divide using long division. Set up for the division P(x) ÷ D(x) the same way as for dividing numbers. x – 4 2x2 – 3x + 4 D(x) P(x) Dividend Divisor
The Long Division Example A. Divide using long division. Set up for the division P(x) ÷ D(x) the same way as for dividing numbers. x – 4 2x2 – 3x + 4 1. Set up the long division D(x) P(x) Dividend Divisor
The Long Division Example A. Divide using long division. Set up for the division P(x) ÷ D(x) the same way as for dividing numbers. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 D(x) P(x) Dividend Divisor
The Long Division Example A. Divide using long division. Set up for the division P(x) ÷ D(x) the same way as for dividing numbers. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 D(x) P(x) Dividend Divisor 2. Enter on top the quotient of the leading terms .
The Long Division Example A. Divide using long division. Set up for the division P(x) ÷ D(x) the same way as for dividing numbers. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2. Enter on top the quotient of the leading terms . 2x D(x) P(x) Dividend Divisor
The Long Division Example A. Divide using long division. Set up for the division P(x) ÷ D(x) the same way as for dividing numbers. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2x 3. Multiply this quotient to the divisor and subtract the result from the dividend. D(x) P(x) Dividend Divisor 2. Enter on top the quotient of the leading terms .
The Long Division Example A. Divide using long division. Set up for the division P(x) ÷ D(x) the same way as for dividing numbers. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2x 2x2 – 8x3. Multiply this quotient to the divisor and subtract the result from the dividend. – ) D(x) P(x) Dividend Divisor 2. Enter on top the quotient of the leading terms .
The Long Division Example A. Divide using long division. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2x 2x2 – 8x3. Multiply this quotient to the divisor and subtract the result from the dividend. – ) – + Set up for the division P(x) ÷ D(x) the same way as for dividing numbers.D(x) P(x) Dividend Divisor 2. Enter on top the quotient of the leading terms . change the signs then add
The Long Division Example A. Divide using long division. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2x 2x2 – 8x3. Multiply this quotient to the divisor and subtract the result from the dividend. – ) – + 5x + 4 Set up for the division P(x) ÷ D(x) the same way as for dividing numbers.D(x) P(x) Dividend Divisor 2. Enter on top the quotient of the leading terms .
The Long Division Example A. Divide using long division. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2x 2x2 – 8x3. Multiply this quotient to the divisor and subtract the result from the dividend. – ) – + 5x + 4 Set up for the division P(x) ÷ D(x) the same way as for dividing numbers.D(x) P(x) Dividend Divisor new dividend 2. Enter on top the quotient of the leading terms . 4. Use this difference from the subtraction as the new dividend, repeat steps 2 and 3.
The Long Division Example A. Divide using long division. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2x + 5 2x2 – 8x3. Multiply this quotient to the divisor and subtract the result from the dividend. 4. Use this difference from the subtraction as the new dividend, repeat steps 2 and 3. – ) – + 5x + 4 Set up for the division P(x) ÷ D(x) the same way as for dividing numbers.D(x) P(x) Dividend Divisor new dividend 2. Enter on top the quotient of the leading terms .
The Long Division Example A. Divide using long division. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2x + 5 2x2 – 8x3. Multiply this quotient to the divisor and subtract the result from the dividend. 4. Use this difference from the subtraction as the new dividend, repeat steps 2 and 3. – ) – + 5x + 4 5x – 20– ) Set up for the division P(x) ÷ D(x) the same way as for dividing numbers.D(x) P(x) Dividend Divisor new dividend 2. Enter on top the quotient of the leading terms .
The Long Division Example A. Divide using long division. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2x + 5 2x2 – 8x3. Multiply this quotient to the divisor and subtract the result from the dividend. 4. Use this difference from the subtraction as the new dividend, repeat steps 2 and 3. – ) – + 5x + 4 5x – 20– ) – + 24 Set up for the division P(x) ÷ D(x) the same way as for dividing numbers.D(x) P(x) Dividend Divisor new dividend 2. Enter on top the quotient of the leading terms .
The Long Division Example A. Divide using long division. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2x + 5 2x2 – 8x3. Multiply this quotient to the divisor and subtract the result from the dividend. 4. Use this difference from the subtraction as the new dividend, repeat steps 2 and 3. 5. Stop when the degree of the new dividend is smaller than the degree of the divisor, i.e. no more quotient is possible. – ) – + 5x + 4 5x – 20– ) – + 24 Set up for the division P(x) ÷ D(x) the same way as for dividing numbers.D(x) P(x) Dividend Divisor new dividend 2. Enter on top the quotient of the leading terms .
The Long Division 2x2 – 3x + 4x – 4 2x + 5 2x2 – 8x 5x + 4 5x – 20 24 Here are the names of the terms: dividend P(x)divisor D(x) quotient Q(x) remainder R(x)
The Long Division 2x2 – 3x + 4x – 4 2x + 5 2x2 – 8x 5x + 4 5x – 20 24 Here are the names of the terms: dividend P(x)divisor D(x) quotient Q(x) remainder R(x) We check easily that x – 4 2x2 – 3x + 4 = 2x + 5 + x – 4 24
The Long Division 2x2 – 3x + 4x – 4 2x + 5 2x2 – 8x 5x + 4 5x – 20 24 Here are the names of the terms: dividend P(x)divisor D(x) quotient Q(x) remainder R(x) We check easily that x – 4 2x2 – 3x + 4 = 2x + 5 + x – 4 24 i.e. = Q + P D R D
The Long Division 2x2 – 3x + 4x – 4 2x + 5 2x2 – 8x 5x + 4 5x – 20 24 Here are the names of the terms: dividend P(x)divisor D(x) quotient Q(x) remainder R(x) We check easily that x – 4 2x2 – 3x + 4 = 2x + 5 + x – 4 24 We summarize the end result from performing the long division algorithm in the following theorem. i.e. = Q + P D R D
The Long Division Long Division Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that
The Long Division Long Division Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x).
The Long Division Long Division Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of Q(x) + D(x) R(x)
The Long Division Long Division Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of x3 – 2x2 + 0x + 3x2 + 1 Q(x) + D(x) R(x)
The Long Division Long Division Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of x3 – 2x2 + 0x + 3x2 + 1 x Q(x) + D(x) R(x)
The Long Division Long Division Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of x3 – 2x2 + 0x + 3x2 + 1 x x3 + x– ) Q(x) + D(x) R(x)
The Long Division Long Division Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of x3 – 2x2 + 0x + 3x2 + 1 x x3 + x– ) –– – 2x2 – x + 3 Q(x) + D(x) R(x)
The Long Division Long Division Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of x3 – 2x2 + 0x + 3x2 + 1 x – 2 x3 + x– ) –– – 2x2 – x + 3 Q(x) + D(x) R(x)
The Long Division Long Division Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of x3 – 2x2 + 0x + 3x2 + 1 x – 2 x3 + x– ) – – ) – – 2x2 – x + 3 Q(x) + D(x) R(x) – 2x2 – 2
The Long Division Long Division Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of x3 – 2x2 + 0x + 3x2 + 1 x – 2 x3 + x– ) – – ) + – x + 5 – – 2x2 – x + 3 – 2x2 – 2Q(x) + D(x) R(x) +
The Long Division Long Division Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of x3 – 2x2 + 0x + 3x2 + 1 x – 2 x3 + x– ) – – ) + – x + 5 – – 2x2 – x + 3 Q(x) + D(x) R(x) + Stop! The degree of R is less than the degree of D – 2x2 – 2
The Long Division Long Division Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of x3 – 2x2 + 0x + 3x2 + 1 x – 2 x3 + x– ) – – ) ++ – x + 5 – – 2x2 – x + 3 Q(x) + D(x) R(x) Hence x2 + 1 x3 – 2x2 + 3 = x – 2 + x2 + 1 – x + 5 – 2x2 – 2
Synthetic Division When the divisor is of the form x – c, we may use a simpler procedure called synthetic division.
Synthetic Division When the divisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4
Synthetic Division When the divisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 1. To set up the synthetic division, list the coefficients of the dividend in descending order,
Synthetic Division When the divisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 2 –3 4 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order,
Synthetic Division When the divisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 2 –3 4 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown.
Synthetic Division When the divisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 2 –3 44 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown.
Synthetic Division When the divisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 2. To "divide", bring down the leading coefficient, 2 –3 44 2 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown.
Synthetic Division When the divisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown. 2. To "divide", bring down the leading coefficient, multiply it by c and enter the result in next column. 2 –3 44 2
Synthetic Division When the divisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown. 2. To "divide", bring down the leading coefficient, multiply it by c and enter the result in next column. 2 –3 44 2 8
Synthetic Division When the divisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown. 2. To "divide", bring down the leading coefficient, multiply it by c and enter the result in next column. Add the column to get the sum. 2 –3 44 2 8 5
Synthetic Division When the divisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown. 2. To "divide", bring down the leading coefficient, multiply it by c and enter the result in next column. Add the column to get the sum. Multiply the sum by c, enter the result in the next column then add. 2 –3 44 2 8 5
Synthetic Division When the divisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown. 2. To "divide", bring down the leading coefficient, multiply it by c and enter the result in next column. Add the column to get the sum. Multiply the sum by c, enter the result in the next column then add. 2 –3 44 2 8 5 20
Synthetic Division When the divisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown. 2 –3 44 2 8 5 202. To "divide", bring down the leading coefficient, multiply it by c and enter the result in next column. Add the column to get the sum. Multiply the sum by c, enter the result in the next column then add. Continue this process to the last column and the procedure stops.
Synthetic Division When the divisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown. 2 –3 44 2 8 5 202. To "divide", bring down the leading coefficient, multiply it by c and enter the result in next column. Add the column to get the sum. Multiply the sum by c, enter the result in the next column then add. Continue this process to the last column and the procedure stops. 24 Add and the procedure stops.
Synthetic Division 2 –3 44 2 8 5 20 The result of the division is read in the bottom row. 24 The last number is the remainder, it’s 24.
Synthetic Division 2 –3 44 2 8 5 20 The result of the division is read in the bottom row. 24 The last number is the remainder, it’s 24. these numbers are the coefficients of the quotient polynomial which is one degree less than the dividend, it’s 2x + 5.
Synthetic Division 2 –3 44 2 8 5 20 The result of the division is read in the bottom row. 24 The last number is the remainder, it’s 24. these numbers are the coefficients of the quotient polynomial which is one degree less than the dividend, it's 2x + 5. Hence x – 4 2x2 – 3x + 4 = 2x + 5 + x – 4 24
Synthetic Division Example C. Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2
Synthetic Division Example C. Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms.
Synthetic Division Example C. Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. 2 0 –7 0 –3 2
Synthetic Division Example C. Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2
Synthetic Division Example C. Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2
Synthetic Division Example C. Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2 –4multiply
Synthetic Division Example C. Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2 –4 add –4
Synthetic Division Example C. Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2 –4multiply –4 8
Synthetic Division Example C. Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2 –4 –4 8 1
Synthetic Division Example C. Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2 –4 –4 8 1 –2 –2
Synthetic Division Example C. Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2 –4 –4 8 1 –2 –2 4 1
Synthetic Division Example C. Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2 –4 –4 8 1 –2 –2 4 1 –2 0
Synthetic Division Example C. Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2 –4 –4 8 1 –2 –2 4 1 –2 0 So x + 2 2x5 – 7x3 – 3x + 2 = 2x4 – 4x3 + x2 – 2x + 1
Synthetic Division Example C. Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2 –4 –4 8 1 –2 –2 4 1 –2 0 So x + 2 2x5 – 7x3 – 3x + 2 = 2x4 – 4x3 + x2 – 2x + 1 Note that because the remainder is 0, we have that 2x5 – 7x3 – 3x + 2 = (x + 2) (2x4 – 4x3 + x2 – 2x + 1) and that x = –2 is a root.
Long Division Exercise A. Divide P(x) ÷ D(x) using long division, D(x) P(x) as Q(x)+ D(x) R(x) with deg R(x) < deg D(x). 1. x + 3 –2x + 3 and write 4. x + 3 x2 – 9 7. x + 3 x2 – 2x + 3 2. x + 1 3x + 2 3. 2x – 1 3x + 1 8. x – 3 2x2 – 2x + 1 9. 2x + 1 –2x2 + 4x + 1 5. x + 2 x2 + 4 6. x – 3 x2 + 9 10. x + 3 x3 – 2x + 3 11. x – 3 2x3 – 2x + 1 12. 2x + 1 –2x3 + 4x + 1 13. x2 + x + 3 x3 – 2x + 3 14. x2 – 3 2x3 – 2x + 1 15. x2 – 2x + 1 –2x3 + 4x + 1 16. x – 1 x30 – 2x20+ 1 16. x + 1 x30 – 2x20 + 1 18. x – 1 xN – 1 (N > 1) (Many of them can be done by synthetic division. )
Synthetic Division B. Divide P(x) ÷ (x – c) using synthetic division, D(x) P(x) as Q(x) + x – c r where r is a number. 1. x + 3 –2x + 3 and write 2. x + 1 3x + 2 3. x – 2 3x + 1 4. x + 3 x2 – 2x + 3 5. x – 3 2x2 – 2x + 1 6. x + 2 –2x2 + 4x + 1 7. x + 3 x3 – 2x + 3 8. x – 3 2x3 – 2x + 1 9. x + 4 –2x3 + 4x + 1 10. x – 1 x30 – 2x + 1 11. x + 1 x30 – 2x20 + 1 12. x – 1 xN – 1 (N > 1) 13. Use synthetic division to verify that (x3 – 7x – 6) / (x + 2) divides completely with remainder 0, then factor x3 – 7x – 6 completely.
The Long Division (Answers to odd problems) Exercise A. 1. x + 3 9 7. 3. 2 3 9. 5. 11. 13. 15. 17. x29 – x28 + x27 – x26 + x25 – x24 + x23 – x22 + x21 – x20 – x19 + x18 – x17 + x16 – x15 + x14 – x13 + x12 – x11 + x10 – x9 + x8 – x7 + x6 – x5 + x4 – x3 + x2 – x2 + 1 –2 + + 2(2x – 1) 5 x + 2 8(x – 2) + x +3 18(x – 5) + (– x + )2 5 – 2(2x – 1) 5 x – 3 49(x2 + 6x +16) + x2 + x + 3 2(2x – 3) (x – 1) – (x – 1)2 5 – 2x –(2x + 4) +
The Long Division Exercise B. 1. 3. 5. 7. 9. 11. x29 – x28 + x27 – x26 + x25 – x24 + x23 – x22 + x21 13. (x + 1)(x + 2)(x – 3) x + 3 9–2 + x – 2 73 + x – 3 13(2x + 4) + x + 3 18(x2 – 3x + 7) – x + 4 113(– 2x2 + 8x – 28) + – x20 – x19 + x18 – x17 + x16 – x15 + x14 – x13 + x12 – x11 + x10 – x9 + x8 – x7 + x6 – x5 + x4 – x3 + x2 – x2 + 1

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3.1 methods of division

  • 1.
  • 2.
    Methods of Division Thequestions of factorability and finding roots of real polynomials (polynomials with real coefficients) are the main themes of this chapter.
  • 3.
    Methods of Division Westart with two division algorithms for polynomials of a single variable x; long division and synthetic division. The questions of factorability and finding roots of real polynomials (polynomials with real coefficients) are the main themes of this chapter.
  • 4.
    Methods of Division Westart with two division algorithms for polynomials of a single variable x; long division and synthetic division. The questions of factorability and finding roots of real polynomials (polynomials with real coefficients) are the main themes of this chapter. Long division of polynomials is analogous to the long division of numbers and it’s a general division method for P(x) ÷ D(x).
  • 5.
    Methods of Division Westart with two division algorithms for polynomials of a single variable x; long division and synthetic division. The questions of factorability and finding roots of real polynomials (polynomials with real coefficients) are the main themes of this chapter. Long division of polynomials is analogous to the long division of numbers and it’s a general division method for P(x) ÷ D(x). Synthetic division is a simpler method for dividing polynomials P(x) by monomials of the form (x – c), i.e. P(x) ÷ (x – c).
  • 6.
    Methods of Division Westart with two division algorithms for polynomials of a single variable x; long division and synthetic division. The questions of factorability and finding roots of real polynomials (polynomials with real coefficients) are the main themes of this chapter. Long division of polynomials is analogous to the long division of numbers and it’s a general division method for P(x) ÷ D(x). Synthetic division is a simpler method for dividing polynomials P(x) by monomials of the form (x – c), i.e. P(x) ÷ (x – c). Synthetic division is particularly useful for checking possible roots or finding remainders of the division.
  • 7.
  • 8.
    The Long Division Setup for the division P(x) ÷ D(x) the same way as for dividing numbers.D(x) P(x) Dividend Divisor
  • 9.
    The Long Division ExampleA. Divide using long division. Set up for the division P(x) ÷ D(x) the same way as for dividing numbers. x – 4 2x2 – 3x + 4 D(x) P(x) Dividend Divisor
  • 10.
    The Long Division ExampleA. Divide using long division. Set up for the division P(x) ÷ D(x) the same way as for dividing numbers. x – 4 2x2 – 3x + 4 1. Set up the long division D(x) P(x) Dividend Divisor
  • 11.
    The Long Division ExampleA. Divide using long division. Set up for the division P(x) ÷ D(x) the same way as for dividing numbers. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 D(x) P(x) Dividend Divisor
  • 12.
    The Long Division ExampleA. Divide using long division. Set up for the division P(x) ÷ D(x) the same way as for dividing numbers. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 D(x) P(x) Dividend Divisor 2. Enter on top the quotient of the leading terms .
  • 13.
    The Long Division ExampleA. Divide using long division. Set up for the division P(x) ÷ D(x) the same way as for dividing numbers. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2. Enter on top the quotient of the leading terms . 2x D(x) P(x) Dividend Divisor
  • 14.
    The Long Division ExampleA. Divide using long division. Set up for the division P(x) ÷ D(x) the same way as for dividing numbers. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2x 3. Multiply this quotient to the divisor and subtract the result from the dividend. D(x) P(x) Dividend Divisor 2. Enter on top the quotient of the leading terms .
  • 15.
    The Long Division ExampleA. Divide using long division. Set up for the division P(x) ÷ D(x) the same way as for dividing numbers. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2x 2x2 – 8x3. Multiply this quotient to the divisor and subtract the result from the dividend. – ) D(x) P(x) Dividend Divisor 2. Enter on top the quotient of the leading terms .
  • 16.
    The Long Division ExampleA. Divide using long division. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2x 2x2 – 8x3. Multiply this quotient to the divisor and subtract the result from the dividend. – ) – + Set up for the division P(x) ÷ D(x) the same way as for dividing numbers.D(x) P(x) Dividend Divisor 2. Enter on top the quotient of the leading terms . change the signs then add
  • 17.
    The Long Division ExampleA. Divide using long division. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2x 2x2 – 8x3. Multiply this quotient to the divisor and subtract the result from the dividend. – ) – + 5x + 4 Set up for the division P(x) ÷ D(x) the same way as for dividing numbers.D(x) P(x) Dividend Divisor 2. Enter on top the quotient of the leading terms .
  • 18.
    The Long Division ExampleA. Divide using long division. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2x 2x2 – 8x3. Multiply this quotient to the divisor and subtract the result from the dividend. – ) – + 5x + 4 Set up for the division P(x) ÷ D(x) the same way as for dividing numbers.D(x) P(x) Dividend Divisor new dividend 2. Enter on top the quotient of the leading terms . 4. Use this difference from the subtraction as the new dividend, repeat steps 2 and 3.
  • 19.
    The Long Division ExampleA. Divide using long division. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2x + 5 2x2 – 8x3. Multiply this quotient to the divisor and subtract the result from the dividend. 4. Use this difference from the subtraction as the new dividend, repeat steps 2 and 3. – ) – + 5x + 4 Set up for the division P(x) ÷ D(x) the same way as for dividing numbers.D(x) P(x) Dividend Divisor new dividend 2. Enter on top the quotient of the leading terms .
  • 20.
    The Long Division ExampleA. Divide using long division. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2x + 5 2x2 – 8x3. Multiply this quotient to the divisor and subtract the result from the dividend. 4. Use this difference from the subtraction as the new dividend, repeat steps 2 and 3. – ) – + 5x + 4 5x – 20– ) Set up for the division P(x) ÷ D(x) the same way as for dividing numbers.D(x) P(x) Dividend Divisor new dividend 2. Enter on top the quotient of the leading terms .
  • 21.
    The Long Division ExampleA. Divide using long division. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2x + 5 2x2 – 8x3. Multiply this quotient to the divisor and subtract the result from the dividend. 4. Use this difference from the subtraction as the new dividend, repeat steps 2 and 3. – ) – + 5x + 4 5x – 20– ) – + 24 Set up for the division P(x) ÷ D(x) the same way as for dividing numbers.D(x) P(x) Dividend Divisor new dividend 2. Enter on top the quotient of the leading terms .
  • 22.
    The Long Division ExampleA. Divide using long division. x – 4 2x2 – 3x + 4 1. Set up the long division 2x2 – 3x + 4x – 4 2x + 5 2x2 – 8x3. Multiply this quotient to the divisor and subtract the result from the dividend. 4. Use this difference from the subtraction as the new dividend, repeat steps 2 and 3. 5. Stop when the degree of the new dividend is smaller than the degree of the divisor, i.e. no more quotient is possible. – ) – + 5x + 4 5x – 20– ) – + 24 Set up for the division P(x) ÷ D(x) the same way as for dividing numbers.D(x) P(x) Dividend Divisor new dividend 2. Enter on top the quotient of the leading terms .
  • 23.
    The Long Division 2x2– 3x + 4x – 4 2x + 5 2x2 – 8x 5x + 4 5x – 20 24 Here are the names of the terms: dividend P(x)divisor D(x) quotient Q(x) remainder R(x)
  • 24.
    The Long Division 2x2– 3x + 4x – 4 2x + 5 2x2 – 8x 5x + 4 5x – 20 24 Here are the names of the terms: dividend P(x)divisor D(x) quotient Q(x) remainder R(x) We check easily that x – 4 2x2 – 3x + 4 = 2x + 5 + x – 4 24
  • 25.
    The Long Division 2x2– 3x + 4x – 4 2x + 5 2x2 – 8x 5x + 4 5x – 20 24 Here are the names of the terms: dividend P(x)divisor D(x) quotient Q(x) remainder R(x) We check easily that x – 4 2x2 – 3x + 4 = 2x + 5 + x – 4 24 i.e. = Q + P D R D
  • 26.
    The Long Division 2x2– 3x + 4x – 4 2x + 5 2x2 – 8x 5x + 4 5x – 20 24 Here are the names of the terms: dividend P(x)divisor D(x) quotient Q(x) remainder R(x) We check easily that x – 4 2x2 – 3x + 4 = 2x + 5 + x – 4 24 We summarize the end result from performing the long division algorithm in the following theorem. i.e. = Q + P D R D
  • 27.
    The Long Division LongDivision Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that
  • 28.
    The Long Division LongDivision Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x).
  • 29.
    The Long Division LongDivision Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of Q(x) + D(x) R(x)
  • 30.
    The Long Division LongDivision Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of x3 – 2x2 + 0x + 3x2 + 1 Q(x) + D(x) R(x)
  • 31.
    The Long Division LongDivision Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of x3 – 2x2 + 0x + 3x2 + 1 x Q(x) + D(x) R(x)
  • 32.
    The Long Division LongDivision Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of x3 – 2x2 + 0x + 3x2 + 1 x x3 + x– ) Q(x) + D(x) R(x)
  • 33.
    The Long Division LongDivision Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of x3 – 2x2 + 0x + 3x2 + 1 x x3 + x– ) –– – 2x2 – x + 3 Q(x) + D(x) R(x)
  • 34.
    The Long Division LongDivision Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of x3 – 2x2 + 0x + 3x2 + 1 x – 2 x3 + x– ) –– – 2x2 – x + 3 Q(x) + D(x) R(x)
  • 35.
    The Long Division LongDivision Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of x3 – 2x2 + 0x + 3x2 + 1 x – 2 x3 + x– ) – – ) – – 2x2 – x + 3 Q(x) + D(x) R(x) – 2x2 – 2
  • 36.
    The Long Division LongDivision Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of x3 – 2x2 + 0x + 3x2 + 1 x – 2 x3 + x– ) – – ) + – x + 5 – – 2x2 – x + 3 – 2x2 – 2Q(x) + D(x) R(x) +
  • 37.
    The Long Division LongDivision Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of x3 – 2x2 + 0x + 3x2 + 1 x – 2 x3 + x– ) – – ) + – x + 5 – – 2x2 – x + 3 Q(x) + D(x) R(x) + Stop! The degree of R is less than the degree of D – 2x2 – 2
  • 38.
    The Long Division LongDivision Theorem: Using long division for P(x) ÷ D(x), we may obtain a quotient Q(x) and a reminder R(x) such that D(x) P(x) = Q(x) + D(x) R(x) with deg R(x) < deg D(x). Example B. Divide x2 + 1 x3 – 2x2 + 3 and write it in the form of x3 – 2x2 + 0x + 3x2 + 1 x – 2 x3 + x– ) – – ) ++ – x + 5 – – 2x2 – x + 3 Q(x) + D(x) R(x) Hence x2 + 1 x3 – 2x2 + 3 = x – 2 + x2 + 1 – x + 5 – 2x2 – 2
  • 39.
    Synthetic Division When thedivisor is of the form x – c, we may use a simpler procedure called synthetic division.
  • 40.
    Synthetic Division When thedivisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4
  • 41.
    Synthetic Division When thedivisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 1. To set up the synthetic division, list the coefficients of the dividend in descending order,
  • 42.
    Synthetic Division When thedivisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 2 –3 4 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order,
  • 43.
    Synthetic Division When thedivisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 2 –3 4 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown.
  • 44.
    Synthetic Division When thedivisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 2 –3 44 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown.
  • 45.
    Synthetic Division When thedivisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 2. To "divide", bring down the leading coefficient, 2 –3 44 2 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown.
  • 46.
    Synthetic Division When thedivisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown. 2. To "divide", bring down the leading coefficient, multiply it by c and enter the result in next column. 2 –3 44 2
  • 47.
    Synthetic Division When thedivisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown. 2. To "divide", bring down the leading coefficient, multiply it by c and enter the result in next column. 2 –3 44 2 8
  • 48.
    Synthetic Division When thedivisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown. 2. To "divide", bring down the leading coefficient, multiply it by c and enter the result in next column. Add the column to get the sum. 2 –3 44 2 8 5
  • 49.
    Synthetic Division When thedivisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown. 2. To "divide", bring down the leading coefficient, multiply it by c and enter the result in next column. Add the column to get the sum. Multiply the sum by c, enter the result in the next column then add. 2 –3 44 2 8 5
  • 50.
    Synthetic Division When thedivisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown. 2. To "divide", bring down the leading coefficient, multiply it by c and enter the result in next column. Add the column to get the sum. Multiply the sum by c, enter the result in the next column then add. 2 –3 44 2 8 5 20
  • 51.
    Synthetic Division When thedivisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown. 2 –3 44 2 8 5 202. To "divide", bring down the leading coefficient, multiply it by c and enter the result in next column. Add the column to get the sum. Multiply the sum by c, enter the result in the next column then add. Continue this process to the last column and the procedure stops.
  • 52.
    Synthetic Division When thedivisor is of the form x – c, we may use a simpler procedure called synthetic division. Example C. Divide using synthetic division. x – 4 2x2 – 3x + 4 1. To set up the synthetic division, list the coefficients of the dividend in descending degree order, put the number c to the left, draw a line below them as shown. 2 –3 44 2 8 5 202. To "divide", bring down the leading coefficient, multiply it by c and enter the result in next column. Add the column to get the sum. Multiply the sum by c, enter the result in the next column then add. Continue this process to the last column and the procedure stops. 24 Add and the procedure stops.
  • 53.
    Synthetic Division 2 –344 2 8 5 20 The result of the division is read in the bottom row. 24 The last number is the remainder, it’s 24.
  • 54.
    Synthetic Division 2 –344 2 8 5 20 The result of the division is read in the bottom row. 24 The last number is the remainder, it’s 24. these numbers are the coefficients of the quotient polynomial which is one degree less than the dividend, it’s 2x + 5.
  • 55.
    Synthetic Division 2 –344 2 8 5 20 The result of the division is read in the bottom row. 24 The last number is the remainder, it’s 24. these numbers are the coefficients of the quotient polynomial which is one degree less than the dividend, it's 2x + 5. Hence x – 4 2x2 – 3x + 4 = 2x + 5 + x – 4 24
  • 56.
    Synthetic Division Example C.Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2
  • 57.
    Synthetic Division Example C.Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms.
  • 58.
    Synthetic Division Example C.Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. 2 0 –7 0 –3 2
  • 59.
    Synthetic Division Example C.Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2
  • 60.
    Synthetic Division Example C.Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2
  • 61.
    Synthetic Division Example C.Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2 –4multiply
  • 62.
    Synthetic Division Example C.Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2 –4 add –4
  • 63.
    Synthetic Division Example C.Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2 –4multiply –4 8
  • 64.
    Synthetic Division Example C.Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2 –4 –4 8 1
  • 65.
    Synthetic Division Example C.Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2 –4 –4 8 1 –2 –2
  • 66.
    Synthetic Division Example C.Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2 –4 –4 8 1 –2 –2 4 1
  • 67.
    Synthetic Division Example C.Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2 –4 –4 8 1 –2 –2 4 1 –2 0
  • 68.
    Synthetic Division Example C.Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2 –4 –4 8 1 –2 –2 4 1 –2 0 So x + 2 2x5 – 7x3 – 3x + 2 = 2x4 – 4x3 + x2 – 2x + 1
  • 69.
    Synthetic Division Example C.Divide using synthetic division. x + 2 2x5 – 7x3 – 3x + 2 Set up the division, make sure to put 0 for the missing x-terms. Since x + 2 = x – (–2), we use x = –2 for the division. 2 0 –7 0 –3 2–2 2 –4 –4 8 1 –2 –2 4 1 –2 0 So x + 2 2x5 – 7x3 – 3x + 2 = 2x4 – 4x3 + x2 – 2x + 1 Note that because the remainder is 0, we have that 2x5 – 7x3 – 3x + 2 = (x + 2) (2x4 – 4x3 + x2 – 2x + 1) and that x = –2 is a root.
  • 70.
    Long Division Exercise A.Divide P(x) ÷ D(x) using long division, D(x) P(x) as Q(x)+ D(x) R(x) with deg R(x) < deg D(x). 1. x + 3 –2x + 3 and write 4. x + 3 x2 – 9 7. x + 3 x2 – 2x + 3 2. x + 1 3x + 2 3. 2x – 1 3x + 1 8. x – 3 2x2 – 2x + 1 9. 2x + 1 –2x2 + 4x + 1 5. x + 2 x2 + 4 6. x – 3 x2 + 9 10. x + 3 x3 – 2x + 3 11. x – 3 2x3 – 2x + 1 12. 2x + 1 –2x3 + 4x + 1 13. x2 + x + 3 x3 – 2x + 3 14. x2 – 3 2x3 – 2x + 1 15. x2 – 2x + 1 –2x3 + 4x + 1 16. x – 1 x30 – 2x20+ 1 16. x + 1 x30 – 2x20 + 1 18. x – 1 xN – 1 (N > 1) (Many of them can be done by synthetic division. )
  • 71.
    Synthetic Division B. DivideP(x) ÷ (x – c) using synthetic division, D(x) P(x) as Q(x) + x – c r where r is a number. 1. x + 3 –2x + 3 and write 2. x + 1 3x + 2 3. x – 2 3x + 1 4. x + 3 x2 – 2x + 3 5. x – 3 2x2 – 2x + 1 6. x + 2 –2x2 + 4x + 1 7. x + 3 x3 – 2x + 3 8. x – 3 2x3 – 2x + 1 9. x + 4 –2x3 + 4x + 1 10. x – 1 x30 – 2x + 1 11. x + 1 x30 – 2x20 + 1 12. x – 1 xN – 1 (N > 1) 13. Use synthetic division to verify that (x3 – 7x – 6) / (x + 2) divides completely with remainder 0, then factor x3 – 7x – 6 completely.
  • 72.
    The Long Division (Answersto odd problems) Exercise A. 1. x + 3 9 7. 3. 2 3 9. 5. 11. 13. 15. 17. x29 – x28 + x27 – x26 + x25 – x24 + x23 – x22 + x21 – x20 – x19 + x18 – x17 + x16 – x15 + x14 – x13 + x12 – x11 + x10 – x9 + x8 – x7 + x6 – x5 + x4 – x3 + x2 – x2 + 1 –2 + + 2(2x – 1) 5 x + 2 8(x – 2) + x +3 18(x – 5) + (– x + )2 5 – 2(2x – 1) 5 x – 3 49(x2 + 6x +16) + x2 + x + 3 2(2x – 3) (x – 1) – (x – 1)2 5 – 2x –(2x + 4) +
  • 73.
    The Long Division ExerciseB. 1. 3. 5. 7. 9. 11. x29 – x28 + x27 – x26 + x25 – x24 + x23 – x22 + x21 13. (x + 1)(x + 2)(x – 3) x + 3 9–2 + x – 2 73 + x – 3 13(2x + 4) + x + 3 18(x2 – 3x + 7) – x + 4 113(– 2x2 + 8x – 28) + – x20 – x19 + x18 – x17 + x16 – x15 + x14 – x13 + x12 – x11 + x10 – x9 + x8 – x7 + x6 – x5 + x4 – x3 + x2 – x2 + 1

Editor's Notes

  • #73 A1 = B1 A17? A17 = B11