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: 58


$$3x^2(x - 2)(x^2 + 2x + 4)$$

Work Step by Step

To factor this polynomial, we need to factor out the greatest common factor of both the coefficient and the variable: The greatest common factor of $3$ and $24$ is $3$. The greatest common factor of $x^5$ and $x^2$ is $x^2$. Therefore, we will factor out $3x^2$ from each term to get: $$3x^2(x^3 - 8)$$ We see that we now have a polynomial that is the difference of two cubes, so we can further factor this polynomial. The formula for factoring the difference of two cubes is given as: $$A^3 - B^3 = (A - B)(A^2 + AB + B^2)$$ We now plug in the values for $A$, which is the $\sqrt[3] x$ or $x$, and $B$, which is the $\sqrt[3] 8$ or $2$, to get: $$3x^2(x - 2)(x^2 + 2x + 2^2)$$ Multiply to simplify: $$3x^2(x - 2)(x^2 + 2x + 4)$$
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