Algebra I : Semester II : Polynomials

Sections:

Introduction  |   Section 1  |  Section 2  |  Section 3  |  Section 4  |  Section 5  |  Section 6

Section Four

Part 1  |  Part 2  |  Part 3  |  Part 4

Algebra 1: Section 4: Polynomials Factoring Using the GCF and by Grouping   Click here for a presentation on Factoring with the GCF Now, let’s review the rules from the presentation.   To Factor Using the GCF Find the GCF of all terms in the polynomial. Write down the GCF and a pair of parenthesis. Figure out the polynomial inside the parenthesis using division or distribution. Check by multiplying the result. Example 1: Factor a Binomial Factor 6xy – 12x2y. 6xy – 12x2y Polynomial GCF = 6xy Find the GCF 6xy – 12x2y =6xy(  ) Write it with a parentheses Divide to find the polynomial 6xy – 12x2y =6xy(1 – 2x) Write it in the parentheses Check: 6xy(1 – 2x) = 6xy(1) + 6xy(-2x)                    = 6xy – 12x2y Example 2: Factor a Trinomial Factor 8a3b – 16a2b2 + 24a. 8a3b – 16a2b2 + 24a Polynomial GCF = 8a Find the GCF 8a3b – 16a2b2 + 24a = 8a(     ) Write it with a parentheses Divide to find the polynomial 8a3b – 16a2b2 + 24a = 8a(a2b – 2ab2 + 3) Write it in the parentheses Check: 8a(a2b – 2ab2 + 3) = 8a(a2b) + 8a(-2ab2) + 8a(3)                               = 8a3b – 16a2b2 + 24a Another way to use the GCF in factoring is to factor by grouping. Use this when the polynomial does not have a common factor, but can be split into pairs of terms that share a common factor. Example 3: Factor by Grouping Factor 5xy + 10x + 4y + 8. This polynomial has no common factor in all four terms. However, if you group the first two terms and the last two terms, they do have a common factor.   5xy + 10x + 4y + 8   = (5xy + 10x) + (4y + 8) Group the first two and last two terms. = 5x(y + 2) + 4(y + 2) Factor the first group and last group. = (y + 2)(5x + 4) They share a common factor of (y + 2), so use the distributive property: ab + ac = a(b + c). Check by using FOIL: This is the same polynomial you started with. Example 4: Factor by Grouping and a Sign Change Factor 35a – 7ab + 4b – 20. 35a – 7ab + 4b – 20   = (35a – 7ab) + (4b – 20) Group terms with common factors. = 7a(5 – b) + 4(b – 5) Factor the GCF of each group. = 7a(5 – b) – 4(5 – b) (b – 5) = -(-b + 5) = -(5 – b) Use the opposite to turn it around. Placing a negative sign in front of the grouping. = (5 – b) (7a – 4) Factor (5 – b) using the distributive property. Check using FOIL. The solution checks out.   To Factor by Grouping Group terms with common factors. Factor the GCF from each group. Factor the common binomial using the distributive property.   Practice Factor each of the following. 20b2 + 10b 10b(2b + 1) 16xy – 24x2 8x(2y – 3x) 14ab – 21bc 7b(2a – 3c) x3 + 9x2 + x x(x2 + 9x + 1) 9a + 6b + 3 3(3a + 2b + 1) 30x2y – 18xy2 + 42xy 6xy(5x – 3y + 7) Factor each by grouping. 12a2 + 9a + 8a + 6 (12a2 + 9a) + (8a + 6) = 3a (4a + 3) + 2(4a + 3) = (4a + 3) (3a + 2) 10x + 14y + 15xz + 21yz (10x + 14y) + (15xz + 21yz) = 2(5x + 7y) + 3z (5x + 7y) = (5x + 7y) (2 + 3z) 6m2 – 8mn – 15m + 20n If you try to factor by grouping using the order that the terms are currently in, you will not get two polynomials in your parenthesis that are exactly the same: 2m(3m – 4n) – 5(3m + 4n) This is why you should re-order the polynomial: 6m2 – 15m – 8mn + 20n = (6m2 – 15m) + (– 8mn + 20n) = 3m (2m – 5) + 4n (-2m + 5) = 3m (2m – 5) – 4n (2m – 5) = (2m – 5) (3m – 4n) 8xy – 12y – 6x + 9 If you try to factor by grouping using the order that the terms are currently in, you will not get two polynomials in your parenthesis that are exactly the same: 4y(2x – 3) – 3(2x + 3) This is why you should re-order the polynomial: 8xy – 6x – 12y + 9 = (8xy – 6x) + (– 12y + 9) = 2x (4y – 3) + 3(-4y + 3) = 2x (4y – 3) – 3(4y – 3) = (4y – 3) (2x – 3) Now go on to the next part

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