Question

factor the following expression.
$7x^4 - 42x^3 + 56x^2$
a. $7x^2(x - 4)(x - 2)$
b. $7x^2(x - 6)(x + 8)$
c. $7x^2(x - 6)(x - 8)$
d. $7x^2(x - 4)(x + 2)$

Explanation:

Step1: Factor out the GCF

First, find the greatest common factor (GCF) of \( 7x^4 \), \( -42x^3 \), and \( 56x^2 \). The GCF of the coefficients \( 7 \), \( -42 \), and \( 56 \) is \( 7 \), and the GCF of the variable terms \( x^4 \), \( x^3 \), and \( x^2 \) is \( x^2 \). So, factor out \( 7x^2 \):
\( 7x^4 - 42x^3 + 56x^2 = 7x^2(x^2 - 6x + 8) \)

Step2: Factor the quadratic

Next, factor the quadratic \( x^2 - 6x + 8 \). We need two numbers that multiply to \( 8 \) and add to \( -6 \). The numbers \( -4 \) and \( -2 \) work because \( (-4) \times (-2) = 8 \) and \( (-4) + (-2) = -6 \). So, \( x^2 - 6x + 8 = (x - 4)(x - 2) \).

Step3: Combine the factors

Substitute the factored quadratic back into the expression:
\( 7x^2(x^2 - 6x + 8) = 7x^2(x - 4)(x - 2) \)

Thus, the factored form of \( 7x^4 - 42x^3 + 56x^2 \) is \( 7x^2(x - 4)(x - 2) \), which corresponds to option A.

Answer:

A. \( 7x^2(x - 4)(x - 2) \)