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Section 3.12

Section 3.12 . Factoring Quadratic Expressions. Questions. What is FOIL? How do we factor ? How do we factor quadratic expressions that do not have leading coefficient = 1? i.e. Reverse FOIL. F.O.I.L. Recall multiplying one binomial by another could be thought of as F.O.I.L.

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Section 3.12

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  1. Section 3.12 Factoring Quadratic Expressions

  2. Questions • What is FOIL? • How do we factor ? • How do we factor quadratic expressions that do not have leading coefficient = 1? i.e. Reverse FOIL

  3. F.O.I.L Recall multiplying one binomial by another could be thought of as F.O.I.L. F irst O utter I nner L ast O F I L

  4. F.O.I.L Recall multiplying one binomial by another could be though of as F.O.I.L. F irst O utter I nner L ast O F I L Factoring trinomials (especially quadratic expressions) is about trying to reverse this process.

  5. Factoring x2 + bx + c This type of binomial, if it factors, will always factor in the form . Because of how F.O.I.L. works, the following must be true about these numbers: So we are looking for the factors of c that sum to b.

  6. For Example To factor 1st we list the factors of -28: {-1, 28}, {1, -28}, {-2, 14}, {2, -14}, {-4, 7}, {4, -7} 2nd we look for which pair adds to +3: {-4, 7} Thus

  7. Exercise 1 Factor each of the following: (a) (b) (c) (d)

  8. Exercise 1 Factor each of the following: (a) (b) (c) (d) (x + 3)(x + 4) (x – 10)(x – 2) (x – 2)(x + 4) (x – 7)(x + 5)

  9. Factoring Quadratics Factoring quadratic expressions whose leading coefficient is not 1 is not as easy in general. Some people can just think their way through F.O.I.L backwards and get the factored form, if this works for you –GREAT! However, if this is not you, we present the box method.

  10. The Box Method • Draw a 2 by 2 box. • Place the squared term and the constant term inside the box in the top left and bottom right respectively. • Find the product of the leading coefficient and the constant term. • Find two numbers whose product is the number from step 3 and whose sum is the coefficient of x from the original expression. • Write these numbers as coefficients of x in the remaining 2 boxes. • Factor out the GCF of each row and column and write them outside the box. • These terms form the factors of the factored form.

  11. For Example We factor 1. 2. Thus product is 2*24 = 48 x -8 2x  -3

  12. Exercise 2 Factor each of the following: (a) (b) (c)

  13. Exercise 2 Factor each of the following: (a) (b) (c) (3x + 2)(x + 5) (7x – 5)(2x + 3) (2x + 5)(2x + 5) = (2x + 5)2

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