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Factoring

Factoring. Learning Goal. I will be able to factor trinomials using factoring by grouping. Special Products. For any numbers a and b: Difference of Squares: a 2 – b 2 = (a + b)(a – b) General Trinomial: acx 2 + (ad + bc)x + bd = (ax + b)(cx + d)

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Factoring

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  1. Factoring

  2. Learning Goal I will be able to factor trinomials using factoring by grouping.

  3. Special Products For any numbers a and b: Difference of Squares: a2 – b2 = (a + b)(a – b) General Trinomial: acx2 + (ad + bc)x + bd = (ax + b)(cx + d) Perfect Square Trinomial: a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2

  4. Factoring By Grouping… Find the factors of step #2 which add up to equal the middle term. (bx) Check for the greatest common factor (GCF). Multiply the first term by the last term. (ax2)(c) = ? Substitute the factors from step #3 into the original trinomial, replacing the middle term. (Make bx two terms) Group the four term polynomial from step #4. (Make 2 binomials.) Factor by Grouping. If a polynomial is unfactorable, it is considered prime.

  5. Factor Completely. If the polynomial is not factorable, write prime. 1. xy – 2x + y2 – 2y 2. x2 + 10x + 9 (xy – 2x) + (y2 – 2y) x(y – 2) + y(y – 2) (y – 2) (x + y) x2 + 9x + x + 9 (x2 + 9x) + (x + 9) x(x + 9) + 1(x + 9) (x + 9) (x + 1)

  6. Factor Completely. If the polynomial is not factorable, write prime. 3. 6x2 + 27x - 15 4. 2x2 – 11x + 12 3[2x2 + 9x – 5] 3[2x2 + 10x – x – 5] 3[(2x2 + 10x) + (-x – 5)] 3[2x(x + 5) – 1(x + 5)] 3(x + 5) (2x – 1) 2x2 – 8x – 3x + 12 (2x2 – 8x) + (-3x + 12) 2x(x – 4) – 3(x – 4) (x – 4) (2x – 3)

  7. Factor Completely. If the polynomial is not factorable, write prime. 5. 3x2 – 17x - 6 6. 6x2 + x – 12 3x2 – 18x + x - 6 (3x2 – 18) + (x – 6) 3x(x – 6) + 1(x – 6) (x – 6) (3x + 1) 6x2 + 9x – 8x – 12 (6x2 + 9x) + (-8x – 12) 3x(2x + 3) – 4(2x + 3) (2x + 3) (3x – 4)

  8. ( + )( - ) y y Factor Completely. If the polynomial is not factorable, write prime. 7. y2 - 9 8. 16x2 + 25y2 y2 – 0y – 9 y2 – 3y + 3y – 9 (y2 – 3y) + (3y – 9) y(y – 3) + 3(y – 3) (y – 3) (y + 3) A Sum of Squares cannot be factored. OR… y2 – 9 (y)2 – (3)2 (y + 3)(y – 3)

  9. Factor Completely. If the polynomial is not factorable, write prime. 9. 2x4 - 2 10. 7mx2 + 2x2n – 7my2 – 2ny2 2(x4 – 1) 2(x2 + 1)(x2 – 1) 2(x2 + 1) (x + 1)(x – 1) (7mx2 + 2x2n) + (-7my2 – 2ny2) x2(7m + 2n) – y2(7m + 2n) (7m + 2n) (x2 – y2) (7m + 2n) (x – y)(x + y)

  10. 4 4 4 Given the perimeter of a square is (12x2 + 32) inches, what is the length of one side in terms of x? P = 4s 12x2 + 32 = 4s 3x2 + 8 = s The length of one side is (3x2 + 8) inches.

  11. The area of a pool is x2 + 8x + 15 yards2. Find the dimensions of the pool in terms of x. A = LW LW = x2 + 8x + 15 LW = x2 + 5x + 3x + 15 LW = (x2 + 5x) + (3x + 15) LW = x(x + 5) + 3(x + 5) LW = (x + 5) (x + 3) The dimensions of the pool are (x + 5) yards by (x + 3) yards.

  12. The area of a triangle is found to be 6x2 + 10x + 4 feet2. Find the height and length of the base for this triangle in terms of x. A = ½bh ½bh = 6x2 + 10x + 4 (2) ½bh = (6x2 + 10x + 4)(2) bh = 12x2 + 20x + 8 bh = 12x2 + 12x + 8x + 8 bh = (12x2 + 12x) + (8x + 8) bh = 12x(x + 1) + 8(x + 1) bh = (x + 1) (12x + 8) The length of the base and the height of the triangle are (x + 1) feet and (12x + 8) feet.

  13. The area of the playground at the new park is 12x2 – 17x + 6 yards2. In order for the parks and recreation committee to put a flower bed along the sides of the park they will need to know the dimensions of the playground. The playground at the new park has dimensions of (4x – 3) yards and (3x – 2) yards. A = LW LW = 12x2 – 17x + 6 LW = 12x2 – 9x – 8x + 6 LW = (12x2 – 9x) + (-8x + 6) LW = 3x(4x – 3) – 2(4x – 3) LW = (4x – 3) (3x – 2)

  14. Betty is trying to wrap her sister’s birthday present. The volume of the box that the present is contained in is 16x4 – 81 units3. What are the dimensions of the box? V = LWH LWH = 16x4 – 81 LWH = (4x2 + 9)(4x2 – 9) LWH = (4x2 + 9) (2x + 3)(2x – 3) The dimensions of the box are (4x2 + 9) units by (2x + 3) units by (2x – 3) units.

  15. x x x x Bobby is looking to put a pool in his backyard. He wants to put a uniform sidewalk around the pool that has a width of x. The area of the space where the pool will go is 12x2 + 57x + 63 feet2. What are the dimensions of the pool without the sidewalk? A = LW LW = 12x2 + 57x + 63 LW = 12x2 + 21x + 36x + 63 LW = (12x2 + 21x) + (36x + 63) LW = 3x(4x + 7) + 9(4x + 7) LW = (4x + 7) (3x + 9) Length of pool: 4x + 7 – 2x = 2x + 7 Width of pool: 3x + 9 – 2x = x + 9 The dimensions of the pool are (2x + 7) feet by (x + 9) feet.

  16. What Did I Learn Today?

  17. Homework Factoring Polynomials WS

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