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Chapter 6: Problem 20

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state. $$32 x-24$$

Short Answer

Expert verified
The greatest common factor of the terms in the polynomial is 8. So, the factored form of the polynomial is \(8(4x - 3)\)

Step by step solution

01

Identify the Greatest Common Factor (GCF)

In the polynomial \(32x - 24\), both terms, 32x and 24, share a common factor of 8.
02

Express the polynomial as a product of the GCF

To express the polynomial as a product of the GCF, divide each term by the GCF. So, \(32x - 24 = 8(4x - 3)\)
03

Check your factorization

To confirm the factorization is accurate, distribute the factor back across the brackets: \(8(4x - 3) = 32x - 24 \). As this statement is true, the factorization \(8(4x - 3)\) is correct.

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Most popular questions from this chapter

Use the \([\mathrm{GRAPH}]\) or \([\mathrm { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$4 x^{2}-12 x+9=(4 x-3)^{2} ;[-5,5,1] \text { by }[0,20,1]$$

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$y^{5}-81 y$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$x^{2}-4 x y-12 y^{2}$$

Use the \([\mathrm{GRAPH}]\) or \([\mathrm { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$\begin{aligned} &x^{4}-16=\left(x^{2}+4\right)(x+2)(x-2) ;[-5,5,1] \text { by }\\\ &[-20,20,2] \end{aligned}$$

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$3 r^{3}-27 r^{2}-210 r$$
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