Question

what is the completely factored form of this polynomial?
$7x^4 + 14x^3 - 168x^2$

a. $7x^3(x + 4)(x - 6)$
b. $7x^2(x + 4)(x - 6)$
c. $7x^2(x - 4)(x + 6)$
d. $7x^3(x - 4)(x + 6)$

Explanation:

Step1: Factor out the GCF

First, find the greatest common factor (GCF) of the terms \(7x^4\), \(14x^3\), and \(-168x^2\). The GCF of the coefficients \(7\), \(14\), and \(-168\) is \(7\), and the GCF of the variable parts \(x^4\), \(x^3\), and \(x^2\) is \(x^2\). So, factor out \(7x^2\):
\(7x^4 + 14x^3 - 168x^2 = 7x^2(x^2 + 2x - 24)\)

Step2: Factor the quadratic

Next, factor the quadratic \(x^2 + 2x - 24\). We need two numbers that multiply to \(-24\) and add to \(2\). Those numbers are \(6\) and \(-4\) (since \(6 \times (-4) = -24\) and \(6 + (-4) = 2\)). So, \(x^2 + 2x - 24 = (x + 6)(x - 4)\)? Wait, no—wait, let's check: \((x + 4)(x - 6) = x^2 - 6x + 4x - 24 = x^2 - 2x - 24\). Wait, no, I made a mistake. Wait, the quadratic is \(x^2 + 2x - 24\). Let's redo: find two numbers \(a\) and \(b\) such that \(a \times b = -24\) and \(a + b = 2\). Let's try \(6\) and \(-4\): \(6 + (-4) = 2\), \(6 \times (-4) = -24\). So, \(x^2 + 2x - 24 = (x + 6)(x - 4)\)? Wait, no, \((x + 6)(x - 4) = x^2 + 6x - 4x - 24 = x^2 + 2x - 24\). Wait, but the original quadratic after factoring out \(7x^2\) is \(x^2 + 2x - 24\). Wait, but let's check the answer options. Wait, option C is \(7x^2(x - 4)(x + 6)\), which is \(7x^2(x + 6)(x - 4)\), which factors to \(7x^2(x^2 + 2x - 24)\), which matches. Wait, but let's check the original polynomial again. Wait, the original polynomial is \(7x^4 + 14x^3 - 168x^2\). Let's expand option C: \(7x^2(x - 4)(x + 6) = 7x^2(x^2 + 6x - 4x - 24) = 7x^2(x^2 + 2x - 24) = 7x^4 + 14x^3 - 168x^2\), which matches. Wait, but let's check the steps again. Wait, when I factored the GCF, I got \(7x^2(x^2 + 2x - 24)\), then factoring \(x^2 + 2x - 24\) gives \((x + 6)(x - 4)\), so the fully factored form is \(7x^2(x - 4)(x + 6)\), which is option C. Wait, but let's check the options again. Option C is \(7x^2(x - 4)(x + 6)\), which is correct. Wait, maybe I made a mistake earlier with the signs. Let's verify:

Expand option C:
\(7x^2(x - 4)(x + 6) = 7x^2[(x)(x) + x(6) - 4(x) - 4(6)] = 7x^2[x^2 + 6x - 4x - 24] = 7x^2[x^2 + 2x - 24] = 7x^4 + 14x^3 - 168x^2\), which matches the original polynomial.

Wait, but let's check the other options. Option B: \(7x^2(x + 4)(x - 6) = 7x^2[x^2 - 6x + 4x - 24] = 7x^2[x^2 - 2x - 24] = 7x^4 - 14x^3 - 168x^2\), which does not match (the middle term is \(-14x^3\) instead of \(+14x^3\)). So option C is correct.

Answer:

C. \(7x^2(x - 4)(x + 6)\)