Polynomial for class 9
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Polynomials can be divided using long division, which produces a quotient and remainder. The remainder must have a degree lower than the divisor or be zero. Therefore, x - 2 cannot be the remainder when dividing a polynomial p(x) by x + 3, since both have degree 1.
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Polynomial for class 9
- 2. Polynomials : An algebraic expression in which the variables involved have only non-negative integral powers is called a polynomial.
- 4. Difference between Algebric Expression and Polynomial: A polynomial is always going to be an algebraic expression, but an algebraic expression doesn't always have to be a polynomial. (i)An algebraic expression is an expression with a variable in it, and a polynomial is an expression with multiple terms with variables in it. Algebraic expression (not polynomial): 3b Polynomial: 4x² + 3x - 7 ii) An expression is not a polynomial if it has a negative exponent or fractional exponent
- 5. Polynomial Can Have • A polynomial can have: Constants Variables Exponents Coefficients
- 6. Degree of Polynomials • The degree of a polynomial is the highest degree for a term. • For e.g.- • The polynomial 3 − 5x + 2x5 − 7x9 has degree 9.
- 7. Types Of Polynomial • Polynomials classified by degree – Degree Name Example undefined Zero 0 0 (Non-zero) Constant 1 1 Linear X+1 2 Quadratic X2+1 3 Cubic X3+1 4 Quartic(Biquadratic) X4+1 5 Quintic X5+1 6 Sextic X6+1 7 Septic X7+1 8 Octic X8+1 9 Nonic X9+1 10 Decic X10+1 100 Hectic X100+1
- 8. Linear Polynomials • In a different usage to the above, a polynomial of degree 1 is said to be linear, because the graph of a function of that form is a line. • For e.g.- • 2x+1 • 11y +3
- 9. Quadratic Polynomials • In mathematics, a quadratic polynomial or quadratic is a polynomial of degree two, also called second-order polynomial. That means the exponents of the polynomial's variables are no larger than 2. • For e.g.- • x2 − 4x + 7 is a quadratic polynomial, while x3 − 4x + 7 is not.
- 10. Cubic Polynomials • Cubic polynomial is a polynomial of having degree of polynomial no more than 3 or highest degree in the polynomial should be 3 and should not be more or less than 3. • For e.g.- • x3 + 11x = 9x2 + 55 • x3+ x2+10x = 20
- 11. Biquadratic Polynomials Biquadratic polynomial is a polynomial of having degree of polynomial is no more than 4 or highest degree in the polynomial is not more or less than 4. For e.g.- 4x4 + 5x3 – x2 + x - 1 9y4 + 56x3 – 6x2 + 9x + 2
- 12. Types Of Polynomial • Polynomial can be classified by number of non-zero term Number of non- zero terms Name Example 0 Zero Polynomial 0 1 Monomial X2 2 Binomial X2+1 3 Trinomial X3 +X+1
- 13. Zero Polynomials • The constant polynomial whose coefficients are all equal to 0. The corresponding polynomial function is the constant function with value 0, also called the zero map. The degree of the zero polynomial is undefined, but many authors conventionally set it equal to -1 or ∞.
- 14. Monomial, Binomial & Trinomial Monomial:- A polynomial with one term. E.g. - 5x3, 8, and 4xy. Binomial:- A polynomial with two terms which are not like terms. E.g. - 2x – 3, 3x5 +8x4, and 2ab – 6a2b5. Trinomial:- A polynomial with three terms which are not like terms. E.g. - x2 + 2x - 3, 3x5 - 8x4 + x3, and a2b + 13x + c.
- 16. Followings are not Polynomial
- 17. 3x4 + 5x2 – 7x + 1 The polynomial above is in standard form. Standard form of a polynomial - means that the degrees of its monomial terms decrease from left to right. term term termterm Polynomial Degree Name using Degree Number of Terms Name using number of terms 7x + 4 1 Linear 2 Binomial 3x2 + 2x + 1 2 Quadratic 3 Trinomial 4x3 3 Cubic 1 Monomial 9x4 + 11x 4 Fourth degree 2 Binomial 5 0 Constant 1 monomial
- 18. State whether each expression is a polynomial. If it is, identify it. 1) 7y - 3x + 4 Trinomial 2) 10x3yz2 Monomial 3) Not a polynomial 2 5 7 2 y y
- 19. The Degree of a monomial is the sum of the exponents of the variables or it is the highest power of one variable polynomial. 1) 5x2 Degree: 2 2) 4a4b3c Degree: 8 3) -3 Degree: 0
- 20. Find the degree of x5 – x3y2 + 4 1. 0 2. 2 3. 3 4. 5 5. 10
- 21. 3) Put in ascending order in terms of y: 12x2y3 - 6x3y2 + 3y - 2x -2x + 3y - 6x3y2 + 12x2y3 4) Put in ascending order: 5a3 - 3 + 2a - a2 -3 + 2a - a2 + 5a3
- 22. Write in ascending order in terms of y: x4 – x3y2 + 4xy – 2x2y3 1. x4 + 4xy – x3y2– 2x2y3 2. – 2x2y3 – x3y2 + 4xy + x4 3. x4 – x3y2– 2x2y3 + 4xy 4. 4xy – 2x2y3 – x3y2 + x4
- 29. Dividing Polynomials Long division of polynomials is similar to long division of whole numbers. dividend = (quotient X divisor) + remainder When you divide two polynomials you can check the answer using the following:
- 30. Division algorithm for polynomials If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find two unique polynomials q(x) and r(x) such that p(x) =g(x) x q(x) + r(x) where r(x) = 0 or degree of r(x) < degree of g(x)
- 31. + 2 231 2 xxx Example: Divide x2 + 3x – 2 by x – 1 and check the answer. x x2 + x 2x – 2 2x + 2 –4 remainder Check: x x x xx 2 2 1. xxxx 2 )1(2. xxxxx 2)()3( 22 3. 2 2 2 x x xx4. 22)1(2 xx5. 4)22()22( xx6. correct(x + 2) quotient (x + 1) divisor + (– 4) remainder = x2 + 3x – 2 dividend Answer: Quotient = x + 2 and Remainder = - 4
- 32. Example: Divide 4x + 2x3 – 1 by 2x – 2 and check the answer. 140222 23 xxxx Write the terms of the dividend in descending order. 2 3 2 2 x x x 1. x2 232 22)22( xxxx 2. 2x3 – 2x2 2233 2)22(2 xxxx 3. 2x2 + 4x x x x 2 2 2 4. + x xxxx 22)22( 2 5. 2x2 – 2x xxxxx 6)22()42( 22 6. 6x – 1 3 2 6 x x7. + 3 66)22(3 xx8. 6x – 6 re m a in d e r5)66()16( xx9. 5 Check: (x2 + x + 3)(2x – 2) + 5 = 4x + 2x3 – 1 Answer: Quotient = x2 + x + 3 Remainder = 5 5 Since there is no x2 term in the dividend, add 0x2 as a placeholder.
- 33. Division of polynomials 33 652 2 xxx x x2 – 2x – 3x + 6 – 3 – 3x + 6 0 Answer: Quotient = x – 3 Remainder = 0 Check: (x – 2)(x – 3) = x2 – 5x + 6 Example: Divide x2 – 5x + 6 by x – 2.
- 34. Example: Divide x3 + 3x2 – 2x + 2 by x + 3 and check the answer. 2233 23 xxxx x2 x3 + 3x2 0x2 – 2x – 2 –2x – 6 8 Check: (x + 3)(x2 – 2) + 8 = x3 + 3x2 – 2x + 2 Answer: Quotient = x2 – 2 Remainder = 8 + 2 Note: the first subtraction eliminated two terms from the dividend. Therefore, the quotient skips a term. + 0x
- 35. Division of polynomials 35 1. Can x - 2 be the remainder on division of a polynomial p(x) by x + 3 ? Ans. No. Here the degree of both the remainder and the divisor are one which is not possible because the remainder is either zero or its degree is lower than that of the degree of the divisor.