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Factoring Trinomials with Negative and Positive Coefficients

This guide provides step-by-step methods for factoring trinomials when the coefficient 'b' is negative and 'c' is positive, as well as when 'b' is positive and 'c' is negative. Using two examples, we explore how to identify factor pairs and organize them effectively for accurate factorization. The importance of considering the signs and possible factorizations is highlighted, along with guided practice problems to reinforce understanding.

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Factoring Trinomials with Negative and Positive Coefficients

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  1. EXAMPLE 1 Factor when b is negative and c is positive Factor 2x2 – 7x + 3. SOLUTION Because bis negative and cis positive, both factors of c must be negative. Make a table to organize your work. You must consider the order of the factors of 3, because the x-terms of the possible factorizations are different.

  2. Factors of 2 Factors of 3 Possible factorization Middle term when multiplied 1,2 –1, –3 (x – 1)(2x – 3) –3x – 2x = –5x 1,2 3, 1 (x – 3)(2x– 1) –x – 6x = –7x Correct 2x2– 7x + 3 = (x – 3)(2x – 1) ANSWER EXAMPLE 1 Factor when b is negative and c is positive

  3. EXAMPLE 2 Factor when b is positive and c is negative Factor 3n2 + 14n – 5. SOLUTION Because b is positive and cis negative, the factors ofc have different signs.

  4. Factors of 3 Factors of –5 Possible factorization Middle term when multiplied 1, 3 1, –5 (n + 1)(3n – 5) –5n + 3n = – 2n 1, 3 –1, 5 (n – 1)(3n + 5) 5n – 3n = 2n 1, 3 5, –1 (n + 5)(3n –1) –n + 15n = 14n Correct 1, 3 –5, 1 (n – 5)(3n + 1) n – 15n = –14n 3n2 + 14n – 5 = (n + 5)(3n – 1) ANSWER EXAMPLE 2 Factor when b is negative and c is positive

  5. ANSWER (t + 2)(3t + 2) ANSWER (s – 1)(4s – 5) ANSWER (h + 7)(2h – 1) for Examples 1 and 2 GUIDED PRACTICE Factor the trinomial. 1. 3t2 + 8t + 4 2. 4s2 – 9s + 5 3.2h2 + 13h – 7

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