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mgb5e_ppt_6_3 - 6.3 Factoring Trinomials of the 2 Form ax...

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§ 6.3 Factoring Trinomials of the Form ax 2 + bx + c and Perfect Square Trinomials
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Martin-Gay, Beginning Algebra, 5ed 2 Factoring Trinomials Returning to the FOIL method, F O I L (3 x + 2)( x + 4) = 3 x 2 + 12 x + 2 x + 8 = 3 x 2 + 14 x + 8 To factor a x 2 + b x + c into (# 1 · x + # 2 )(# 3 · x + # 4 ), note that a is the product of the two first coefficients, c is the product of the two last coefficients and b is the sum of the products of the outside coefficients and inside coefficients. Note that b is the sum of 2 products, not just 2 numbers, as in the last section.
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Martin-Gay, Beginning Algebra, 5ed 3 Factor the polynomial 21 x 2 – 41 x + 10. Possible factors of 21 x 2 are { x , 21 x } or {3 x , 7 x }. Since the middle term is negative, possible factors of 10 must both be negative: {-1, -10} or {-2, -5}. We need to methodically try each pair of factors until we find a combination that works, or exhaust all of our possible pairs of factors. Factoring Polynomials Continued. Example:
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Martin-Gay, Beginning Algebra, 5ed 4 We will be looking for a combination that gives the sum of the products of the outside terms and the inside terms equal to - 41 x . Factors of 21 x 2 Resulting Binomials Product of Outside Terms Product of Inside Terms Sum of Products Factors of 10 { x , 21 x }{1, 10}( x 1 )( 21 x 10 ) –10 x - 21 x – 31 x ( x 10 )( 21 x 1 ) x - 210 x – 211 x { x , 21 x } {2, 5} ( x 2 )( 21 x 5 ) –5 x - 42 x – 47 x ( x 5 )( 21 x 2 ) –2 x - 105 x – 107 x Factoring Polynomials Continued. Example continued:
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Martin-Gay, Beginning Algebra, 5ed 5 Factors of 21 x 2 Resulting Binomials Product of Outside Terms Product of Inside Terms Sum of Products Factors of 10 ( 3 x 5 )( 7 x 2 ) - 6 x - 35 x - 41 x {3 x , 7 x }{1, 10}( 3 x 1 )( 7 x 10 ) - 30 x - 7 x - 37 x ( 3 x 10 )( 7 x 1 ) - 3 x - 70 x - 73 x {3 x , 7 x } {2, 5} ( 3 x 2 )( 7 x 5 ) - 15 x - 14 x - 29 x Factoring Polynomials Continued. Example continued:
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Martin-Gay, Beginning Algebra, 5ed 6 Check the resulting factorization using the FOIL method.
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