Question

one of the factors of ( h(x) = 56x^2 - 13x - 3 ) is ( (7x + 1) ). what is another factor?
a ( 8x - 3 )
b ( 8x + 4 )
c ( 8x - 4 )
d ( 8x + 3 )

Explanation:

Step1: Recall polynomial factoring

To find the other factor of \( h(x) = 56x^2 - 13x - 3 \) given one factor \( (7x + 1) \), we can use polynomial division or factor by grouping. Let's use polynomial division. Divide \( 56x^2 - 13x - 3 \) by \( 7x + 1 \).

Step2: Perform polynomial long division

Divide the leading term \( 56x^2 \) by \( 7x \) to get \( 8x \). Multiply \( (7x + 1) \) by \( 8x \) to get \( 56x^2 + 8x \). Subtract this from \( 56x^2 - 13x - 3 \):
\[

$$\begin{align*} (56x^2 - 13x - 3) - (56x^2 + 8x) &= 56x^2 - 13x - 3 - 56x^2 - 8x\\ &= -21x - 3 \end{align*}$$

\]
Now divide the leading term \( -21x \) by \( 7x \) to get \( -3 \). Multiply \( (7x + 1) \) by \( -3 \) to get \( -21x - 3 \). Subtract this from \( -21x - 3 \):
\[
(-21x - 3) - (-21x - 3) = 0
\]
So the quotient is \( 8x - 3 \), which is the other factor.

Answer:

A. \( 8x - 3 \)