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Algebra I Notes Section 9.5 (A) Factoring x 2 + bx + c With Leading Coefficient = 1

Algebra I Notes Section 9.5 (A) Factoring x 2 + bx + c With Leading Coefficient = 1 To factor a quadratic expression means to write it as a product of ________ or _______ linear factors.

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Algebra I Notes Section 9.5 (A) Factoring x 2 + bx + c With Leading Coefficient = 1

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  1. Algebra I Notes Section 9.5 (A) Factoring x2 + bx + c With Leading Coefficient = 1 To factor a quadratic expression means to write it as a product of ________ or _______ linear factors. In this section, we will learn how to factor a quadratic trinomial with a leading coefficient of _____. We know from the FOIL method that: (x + p)(x + q) = x2 + (p + q)x+ (pq)c So to factorx2 + bx + c, we need to find numberspandqsuch that: p+ q = b (Linear coefficient)AND pq= c (Constant Term) Example:bc x2 + 6x + 8 p + q = b pq = c 4 + 2 = 6 (4)(2) = 8 Answer: (x + 4)(x + 2) factors 2 1

  2. ** P & Q are factors of the Constant Term and they add up to the Coefficient of the Linear Term. Example – Factoringwhen b and c are positive 1.x2+ 3x + 2 1st: Identify b and cb = ______ c= ______ 2nd: Find two numbers whose sum is ____ and productis ______ x2+ 3x + 2 = (x + p)(x + q) = 3rd: Use FOIL to check Example– Factoring when b is negative and c is positive 2.x2– 5x + 6 1st: Identify b and cb = ______ c= ______ 2nd: Find two numbers whose sum is _______ and productis ________ x2– 5x + 6 = (x + p)(x + q) pand q must be negativenumbers =3rd: Use FOIL to check 3 2 3 2 (x + 1)(x + 2) (x + 1)(x + 2) x2 + 2x + x + 2 x2 + 3x + 2 -5 6 -5 6 (x – 2)(x – 3) (x – 2)(x – 3) x2 - 3x – 2x + 6 x2 – 5x + 6

  3. Example– Factoring when b and c are negative. 3. x2 – 2x – 8 1st: Identify b and cb= _____ c= _______ 2nd: Find two numbers whose sum is ______ and productis ________ x2– 2x – 8 = (x + p)(x + q) pand q can’t both be negative numbers = 3rd: Use FOIL to check -2 -8 -2 -8 (x – 4)(x + 2) (x – 4)(x + 2) x2 + 2x – 4x – 8 x2 – 2x - 8 Example– Factoring when b is positive and c is negative. 4.x2 + 7x – 18 1st: Identify b and cb= _______ c= _______ 2nd: Find two numbers whose sum is _____ and productis ________ x2+ 7x – 18 = (x + p)(x + q) pand q can’t both be negative numbers = 3rd: Use FOIL to check 7 -18 7 -18 (x + 9)(x – 2) (x + 9)(x – 2) x2 – 2x + 9x – 18 x2 + 7x - 18

  4. More Examples: Factor the following quadratic trinomials. 5. x2 – 8x – 9 6. x2 + 17x – 60 7. 48 + 13x - x2 8. 72 + 22x + x2 Sum of factors = -8 Product of factors = -9 Sum of factors = 17 Product of factors = -60 (x – 9)(x + 1) (x + 20)(x – 3) -x2 + 13x + 48 x2 + 22x + 72 -1(x2 – 13x – 48) Sum of factors = 22 Product of factors = 72 Sum of factors = -13 Product of factors = -48 (x + 18)(x + 4) -(x – 16)(x + 3)

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