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§8.8 Factoring by Grouping

§8.8 Factoring by Grouping. Check: 6 x 3 + 3 x 2 – 4 x – 2 (2 x + 1)(3 x 2 – 2). = 6 x 3 – 4 x + 3 x 2 – 2 Use FOIL. = 6 x 3 + 3 x 2 – 4 x – 2 Write in standard form. Example 1: Factoring a Cubic Polynomial. Factor 6 x 3 + 3 x 2 – 4 x – 2.

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§8.8 Factoring by Grouping

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  1. §8.8 Factoring by Grouping

  2. Check: 6x3 + 3x2 – 4x – 2 (2x + 1)(3x2 – 2) = 6x3 – 4x + 3x2 – 2 Use FOIL. = 6x3 + 3x2 – 4x – 2 Write in standard form. Example 1: Factoring a Cubic Polynomial Factor 6x3 + 3x2 – 4x – 2. 6x3 + 3x2 – 4x – 2 = 3x2(2x + 1) – 2(2x + 1) Factor the GCF from each group of two terms. = (2x + 1)(3x2 – 2) Factor out (2x + 1).

  3. Example 2: Factoring a Polynomial Completely Factor 8t4 + 12t3 + 16t + 24. 8t4 + 12t3 + 16t + 24 = 4(2t4 + 3t3 + 4t + 6) Factor out the GCF, 4. = 4[t3(2t + 3) + 2(2t + 3)] Factor by grouping. = 4(2t + 3)(t3 + 2) Rewrite factors as two binomials.

  4. Find two factors of ac that have a sum b. Use mental math to determine a good place to start. Step 3:Factors Sum –2(18) = –36 –2 + 18 = 16 –3(12) = –36 –3 + 12 = 9 –4(9) = –36 –4 + 9 = 5 Example Ω: Factoring a Trinomial by Grouping Factor 24h2 + 10h – 6. Step 1:24h2 + 10h – 6 = 2(12h2 + 5h – 3)   Factor out the GCF, 2. Step 2: 12 • –3 = –36 Find the product of ac.

  5. Example Ω: Factoring a Trinomial by Grouping Step 4:12h2– 4h+ 9h – 3 Rewrite the trinomial. Step 5:4h(3h– 1) + 3(3h – 1) Factor by grouping. (4h + 3)(3h – 1) Rewrite factors as two binomials. 24h2 + 10h – 6 = 2(4h + 3)(3h – 1)Include the GCF in your final answer.

  6. Example 3: Finding the Dimensions of a Rectangular Prism A rectangular prism has a volume of 36x3 + 51x2 + 18x. Factor to find the possible expressions for the length, width, and height of the prism. Factor 36x3 + 51x2 + 18x. Step 1: 3x(12x2 + 17x + 6) Factor out the GCF, 3x. Step 2: 12 • 6 = 72 Find the product of ac.

  7. Find two factors of ac that have sum b. Use mental math to determine a good place to start. Step 3:Factors  Sum 4 • 18 4 + 18 = 22 6 • 12 6 + 12 = 18 8 • 9 8 + 9 = 17 Example 3: Finding the Dimensions of a Rectangular Prism A rectangular prism has a volume of 36x3 + 51x2 + 18x. Factor to find the possible expressions for the length, width, and height of the prism. Factor 36x3 + 51x2 + 18x.

  8. Example 3: Finding the Dimensions of a Rectangular Prism Step 4: 3x(12x2 + 8x + 9x + 6) Rewrite the trinomial. Step 5: 3x[4x(3x + 2) + 3(3x + 2)] Factor by grouping. 3x(4x + 3)(3x + 2) Rewrite the factors as binomials. The possible dimensions of the prism are: 3x, (4x + 3), and (3x + 2).

  9. Summary of Factoring: Factoring Polynomials 1.) Factor out the greatest common factor (GCF). 2.) If the polynomial has two terms or three terms, look for a difference of two squares, a product of two squares, or a pair of binomial factors. 3.) If there are four or more terms, group terms and factor to find common binomial factors. 4.) As a final check, make sure there are no common factors other than 1. Summary

  10. Assignment: Pg. 531-532 9-33 Left

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