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- 1. Lesson 2 Factoring by Grouping and Factoring General Trinomials
- 2. Factoring By Grouping Example: Factor 𝑥2 − 2𝑥 + 𝑥𝑦 − 2𝑦 completely Solution: 𝑥2 − 2𝑥 + 𝑥𝑦 − 2𝑦 = 𝑥2 − 2𝑥 Group the terms = 𝑥 𝑥 − 2 + 𝑦 𝑥 − 2 Factor out x and y = 𝑥 − 2 𝑥 + 𝑦 Factor. Some Polynomials may contain common monomial or binomial factors. When a polynomial has four or more terms, common factors can sometimes be used in factoring by grouping. The terms are grouped such that the common factor in a group is also the common factor in the other group.
- 3. Factoring by Grouping and Factoring General Trinomial Example: Factor 2𝑥𝑦 + 𝑥2 𝑦 − 6 − 3𝑥. Solution: 2𝑥𝑦 + 𝑥2 𝑦 − 6 − 3𝑥 = 2𝑥𝑦 + 𝑥2 𝑦 − 6 + 3𝑥 Recall that −6 − 3𝑥 = − 6 + 3𝑥 . = 𝑥𝑦 2 + 𝑥 − 3 2 + 𝑥 = (2 + 𝑥)(𝑥𝑦 − 3)
- 4. Factoring by Grouping and Factoring General Trinomial Example: Factor 4𝑥𝑦 − 𝑥𝑧 + 4𝑥𝑤 − 𝑦𝑧 completely Solution: 4𝑥𝑦 − 𝑤𝑧 + 4𝑥𝑤 − 𝑦𝑧 = 4𝑥𝑦 + 4𝑥𝑤 − 𝑦𝑧 + 𝑤𝑧 rearrange the terms; 4𝑥𝑦 and 𝑤𝑧 have no common factor. = 4𝑥 𝑦 = 𝑤 − 𝑧(𝑦 + 𝑤) At times, the basic approach to be used to factor can be recognized only after rearranging and regrouping the terms
- 5. Factoring by Grouping and Factoring General Trinomial Example: Factor 4𝑥𝑦 − 𝑥𝑧 + 4𝑥𝑤 − 𝑦𝑧 completely Solution: 4𝑥𝑦 − 𝑤𝑧 + 4𝑥𝑤 − 𝑦𝑧 = 4𝑥𝑦 + 4𝑥𝑤 − 𝑦𝑧 + 𝑤𝑧 rearrange the terms; 4𝑥𝑦 and 𝑤𝑧 have no common factor. = 4𝑥 𝑦 = 𝑤 − 𝑧(𝑦 + 𝑤)
- 6. Use these steps when factoring four or more terms by grouping: 1. Rearrange and group terms. Collect the terms into two groups so that each group has a common factor. 2. Factor within a group. Remove the common factor from each group. 3. Factor the entire polynomial. Remove the common binomial factor from the entire polynomial. 4. Write the expression as a product of the common factor and the remaining factor.
- 7. Factoring 𝑥2 + 𝑏𝑥 + 𝑐 Example: Factor 𝑥2 + 5𝑥_6 completely. Solution: 𝑥2 + 5𝑥 + 6 The factors are of the form (𝑥+ )(𝑥+ ). We need two numbers whose product is 6 and whose sum is 5. To check your answer, use the FOIL method to multiply the factors. 𝑥 + 2 𝑥 + 3 = 𝑥2 + 3𝑥 + 2𝑥 + 6 = 𝑥2 + 5𝑥 + 6 Factoring a trinomial of the form 𝑥2 + 𝑏𝑥 + 𝑐 means to express the trinomial as the product of two binomials of the form 𝑥 + 𝑚 ( 𝑥 + Factors of 6 Product Sum 1,6 (1)(6)=6 1+6=7 2,3 (2)(3)=6 2+3=5
- 8. Factoring by Grouping 𝑎𝑥2 + 𝑏𝑥 + 𝑐 𝑤ℎ𝑒𝑟𝑒 𝑎 ≠ 1 Example: Factor 3𝑥2 + 5𝑥 + 2 Solution: Method 1:Trial and Error a. Factor the first and last terms. 3𝑥2 → 3𝑥 ∙ 𝑥 2 → 2 ∙ 1 b. Write the possible pairs of factor. 3𝑥 + 2 𝑥 + 1 (3𝑥 + 1)(𝑥 + 2) c. Check for the correct middle term using FOIL method. 3𝑥 + 1 𝑥 + 2 ⇒ 6𝑥 + 𝑥 = 7𝑥 Wrong middle term 3𝑥 + 2 𝑥 + 1 ⇒ 3𝑥 + 2𝑥 = 5𝑥 Correct middle term d. Answer: (3𝑥 + 2)(𝑥 + 1) When the coefficient 𝑎 of the term 𝑎𝑥2 is not 1, there are more possibilities for factoring to consider. We will consider two methods of factoring this trinomial using trial and error, and factoring by grouping.
- 9. Method 2:Using Factoring By Grouping 3𝑥2 + 5𝑥 + 2 a. Get the product of the first and last terms. 3𝑥2 2 = 6𝑥2 b. Find two numbers whose products is 6𝑥2 and whose sum is 5𝑥. The numbers are 2𝑥 𝑎𝑛𝑑 3𝑥. c. Arrange the terms and apply factoring by grouping. 3𝑥2 + 5𝑥 + 2 = 3𝑥2 + 2𝑥 + 3𝑥 + 2 = 3𝑥2 + 2𝑥 + 3𝑥 + 2 = 𝑥 3𝑥 + 2 + 1 3𝑥 + 2 = (3𝑥 + 2)(𝑥 + 1) Factors of 𝟔𝒙 𝟐 Product Sum 6𝑥, 𝑥 6𝑥 𝑥 = 6𝑥2 6𝑥 + 𝑥 = 7𝑥 2𝑥, 3𝑥 2𝑥 3𝑥 = 6𝑥2 2𝑥 + 3𝑥 = 5𝑥
- 10. Here are the steps in factoring a trinomial in the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐, where 𝑎 ≠ 1using the trial and error method: 1. Look for a common factor, if any. 2. Factor the first term 𝑎𝑥2 3. Factor the last term c. 4. Look for the factors of 𝑎𝑥2 and such that the sum of their product is the middle term 𝑏𝑥. The Following are the steps in factoring trinomial in the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐, 𝑤ℎ𝑒𝑟𝑒 𝑎 ≠ 1using the grouping method: 1. Look for a common factor, if any. 2. Get the product of the first and last terms, then test if it is possible to find factors whose sum is equal to the middle term. 3. Think of two expressions whose product is equal to (𝑎𝑥2 )(c)and whose sum is equal to 𝑏𝑥. 4. Arrange the terms and apply the basic steps in factoring by grouping.
- 11. Remember the steps in factoring by grouping are as follows: 1. Rearrange and group terms. 2. Factor within s group. 3. Factor the entire polynomial. 4. Write the expression as a product of the common factor and remaining factor.