Question

a function and one of its factors is given. use synthetic division to determine another factor.
$k(x) = 4x^3 + 7x^2 - 11x - 5$; $4x - 5$
a $-4x^2 - 12x - 4$
b $-4x^2 + 12x + 4$
c $4x^2 - 12x + 4$
d $4x^2 + 12x + 4$

Explanation:

Step1: Find the root for the factor \(4x - 5\)

Set \(4x - 5 = 0\), then \(x=\frac{5}{4}\).

Step2: Set up synthetic division

The coefficients of \(k(x)=4x^{3}+7x^{2}-11x - 5\) are \(4,7,-11,-5\). We use synthetic division with root \(\frac{5}{4}\):
\[

$$\begin{array}{r|rrrr} \frac{5}{4} & 4 & 7 & -11 & -5 \\ & & 5 & 15 & 5 \\ \hline & 4 & 12 & 4 & 0 \\ \end{array}$$

\]
The numbers at the bottom (excluding the last zero) are the coefficients of the quotient polynomial. So the quotient is \(4x^{2}+12x + 4\).

Answer:

D. \(4x^{2}+12x + 4\)