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Chapter 6: Problem 51

Factor each polynomial using the negative of the greatest common factor. $$-8 x^{4}+32 x^{3}+16 x^{2}$$

Short Answer

Expert verified
The factored form of the polynomial using the negative of the greatest common factor is \( -8x^{2} (x^{2} - 4x - 2) \).

Step by step solution

01

Identifying the Greatest Common Factor

First, let's find the greatest common factor (GCF) of all the terms in the polynomial. For the polynomial \( -8x^{4} + 32x^{3} + 16x^{2} \), the GCF of the coefficients is 8 and the GCF of the variables is \( x^{2} \) as it is the largest power of x that divides all terms in the polynomial.
02

Using Negative of the GCF to Factor the Polynomial

Next, factor the polynomial using the GCF, which is \( -8x^{2} \). So the polynomial \(-8x^{4} + 32x^{3} + 16x^{2}\) can be factored as \( -8x^{2} (x^{2} - 4x - 2)\) After pulling out the GCF, each term in the parentheses is obtained by dividing the corresponding term in the initial polynomial by the GCF.
03

Checking the Factored Form

To confirm if the factored form is correct, distribute \( -8x^{2} \) across each term in the parentheses to ensure that you obtain the original polynomial. When you distribute the \( -8x^{2} \), you will find that you end up with the original polynomial: \(-8x^{4} + 32x^{3} + 16x^{2}\)

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Most popular questions from this chapter

Factor completely. $$4 x^{4}-9 x^{2}+5$$

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