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Algebra I Notes Section 9.6 (A) Factoring ax 2 + bx + c With Leading Coefficient ≠ 1

Algebra I Notes Section 9.6 (A) Factoring ax 2 + bx + c With Leading Coefficient ≠ 1. In section 9.5, we learned how to factor quadratic polynomials whose leading coefficient = 1. We will now learn how to factor quadratic polynomials whose leading coefficient is ≠ 1.

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Algebra I Notes Section 9.6 (A) Factoring ax 2 + bx + c With Leading Coefficient ≠ 1

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  1. Algebra I NotesSection 9.6 (A) Factoring ax2 + bx + c With Leading Coefficient ≠ 1

  2. In section 9.5, we learned how to factor quadratic polynomials whose leading coefficient = 1. We will now learn how to factor quadratic polynomials whose leading coefficient is ≠ 1. To factor quadratic polynomials when a ≠ 1, we need to find the factors of a and the factors of c such that when we add the Outer and Inner products of the FOIL method, their sum = b. Example – Factor the polynomials We must find the factors of a and c 1. 3x2 – 17x + 10 - the factors of 3 (or a) are: ___________ - the factors of 10 (or c) are : ___________ We know that the binomials (without their signs) must be one of these four forms because of the factors: 1. 2. 3. 4. 1 , 3 1, 10 & 2, 5 (x 1)(3x 10) (x 10)(3x 1) (x 2)(3x 5) (x 5)(3x 2)

  3. If we look at the FOIL method for each of the possible factors, we get: FO I L 1. +++ 2. +++ 3. +++ 4. +++ We know that in the original polynomial3x2 – 17x + 10, 10 (or c) must be positive, so the L multiplication of the FOIL method must yield a + 10 (for c). At the same time, the sum of the O and I multiplication must yield a - 17 (for b). With the correct choice of signs, which sum (of O and I) can possibly add up to -17? _______________ So we conclude: 3x2 – 17x + 10 = ____________________ 3x2 10x 3x 10 3x2 x 30x 10 3x2 5x 6x 10 3x2 2x 15x 10 -2x & -15x (x – 5)(3x – 2)

  4. 1, 3 2. 3x2– 4x – 7 - the factors of 3 (or a) are: _____________ - the factors of 7 (or c) are : _____________ We know that the binomials (without their signs) must be one of these two forms because of the factors: 1.2. If we look at the FOIL method for each of the possible factors, we get: FO I L 1. + ++ 2. + + + We know that in the original polynomial3x2 – 4x – 7, -7 (or c) must be negative, so the L multiplication of the FOIL method must yield a -7 (for c). At the same time, the sum of the O and I multiplication must yield a - 4 (for b). With the correct choice of signs, which sum (of O and I) can possibly add up to -4? ________________ So we conclude: 3x2 – 4x – 7 = ________________ 1, 7 (x 1)(3x 7) (x 7)(3x 1) 7x 3x x 21x -7x & 3x (x + 1)(3x – 7)

  5. 2 3. 6x2– 2x – 8 - begin by factoring out the common factor ___________ - the factors of 3 (or a) are: _______________ - the factors of 4 (or c) are : _______________ We know that the binomials (without their signs) must be one of these three forms because of the factors: 1. 2. 3. If we look at the FOIL method for each of the possible factors, we get: FOIL 1. + ++ 2. + ++ 3. + ++ We know that in the FACTORED polynomial3x2 – x – 4, -4 (or c) must be negative, so the L multiplication of the FOIL method must yield a -4 (for c). At the same time, the sum of the O and I multiplication must yield a - 1 (for b). 1, 3 2(3x2 – x – 4) 1, 4 & 2, 2 (x 1)(3x 4) (x 4)(3x 1) (x 2)(3x 2) 4x 3x x 12x 2x 6x With the correct choice of signs, which sum (of O and I) can possibly add up to -1? ________________ So we conclude: 2(3x2 – x – 4) = _______________________ -4x & 3x 2(x + 1)(3x – 4)

  6. More Examples - Factor the polynomials. 1. 3t2 + 16t + 5 2. 6b2– 11b – 2 Factors of a : 1, 3 Factors of a : 1, 6 2, 3 Factors of c : 1, 5 Factors of c : 1, 2 (t 1)(3t 5) (t 5)(3t 1) (2b 1)(3b 2) (2b 2)(3b 1) 5t , 3t t , 15t 4b , 3b 2b , 6b Must add to 16t : (b 1)(6b 2) (b 2)(6b 1) 2b , 6b b , 12b (t + 5)(3t + 1) Must add to -11b : (b – 2)(6b + 1)

  7. 3. 5w2 – 9w – 2 4. 4x2– 6x – 4 Factor out GCF : 2 Factors of a : 1, 5 2(2x2 – 3x – 2) Factors of c : 1, 2 Factors of a : 1, 2 (w 1)(5w 2) (w 2)(5w 1) Factors of c : 1, 2 2w , 5w w , 10w (x 1)(2x 2) (x 2)(2x 1) Must add up to -9w : 2x , 2x x , 4x Must add up to -3x : (w - 2)(5w + 1) 2(x – 2)(2x + 1)

  8. 5. 12y2 – 22y – 20 6. 9x2 + 21x – 18 Factor out GCF : 2 Factor out GCF : 3 2(6y2 – 11y – 10) 3(3x2 + 7x – 6) Factors of a : 1, 6 2, 3 Factors of a : 1, 3 Factors of c : 1, 6 2, 3 Factors of c : 1, 10 2, 5 (2y 2)(3y 5) (2y 5)(3y 2) (x 2)(3x 3) (x 3)(3x 2) 10y , 6y 4y , 15y 3x , 6x 2x , 9x Must add up to -11y : Must add up to 7x : 2(2y – 5)(3y + 2) 3(x + 3)(3x – 2)

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