Introductory Algebra for College Students (7th Edition)

Published by Pearson
ISBN 10: 0-13417-805-X
ISBN 13: 978-0-13417-805-9

Chapter 6 - Review Exercises - Page 480: 60


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

Work Step by Step

To factor this polynomial, we need to factor out the greatest common factor of both the coefficients and the constant: The greatest common factor of the coefficients and constant $5$, $20$, and $105$ is $5$. Therefore, we will factor out $5$ from each term to get: $$5(x^2 + 4x - 21)$$ We see that we now have a polynomial that we can factor even further. We want to find out which combination of factors for $-21$ will add up to $4$. Let us look at the factors for $-21$: $(-21)$ and $(1)$ or $(21)$ and $(-1)$ $(-7)$ and $(3)$ or $(7)$ and $(-3)$ We want a bigger positive number and a smaller negative factor so that their addition will give a positive number, so we should choose $7$ and $-3$: $$5(x + 7)(x - 3)$$ If we want to check if we had factored correctly, we multiply the binomials using the FOIL method to see if we get the same original equation: $$5[(x)(x) - 3(x) + 7(x) + (7)(-3)]$$ Multiplying the terms, we get: $$5[(x)(x) - 3(x) + 7(x) + (7)(-3)]$$ $$5(x^2 - 3x + 7x - 21)$$ Combine like terms: $$5(x^2 + 4x - 21)$$ Multiply $5$ with each of the terms of the trinomial: $$5x^2 + 20x - 105$$ We see that we get the same original equation, so we know that our factoring was correct.
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