Chapter 3 - Exponents, Polynomials and Functions - 3.5 Special Factoring Techniques - 3.5 Exercises - Page 280: 55

$(5x-3)(4x+7)$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To factor the given expression, $20x^2+23x-21 ,$ factor first the $GCF.$ Then find two numbers whose product is $ac$ and whose sum is $b$ in the quadratic expression $ax^2+bx+c.$ Use these $2$ numbers to decompose the middle term of the given quadratic expression and then use factoring by grouping. $\bf{\text{Solution Details:}}$ Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $20(-21)=-240$ and the value of $b$ is $23 .$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{ -12,35 \right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to \begin{array}{l}\require{cancel} 20x^2-12x+35x-21 .\end{array} Grouping the first and second terms and the third and fourth terms, the given expression is equivalent to \begin{array}{l}\require{cancel} (20x^2-12x)+(35x-21) .\end{array} Factoring the $GCF$ in each group results to \begin{array}{l}\require{cancel} 4x(5x-3)+7(5x-3) .\end{array} Factoring the $GCF= (5x-3)$ of the entire expression above results to \begin{array}{l}\require{cancel} (5x-3)(4x+7) .\end{array}

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