Problem 51 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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: \(-8x^{4} + 32x^{3} + 16x^{2}\)