Question

question
use synthetic division to find the result when $4x^3 + 10x^2 - 25x - 4$ is divided by $x + 4$.

Explanation:

Step1: Identify the root for synthetic division

For the divisor \(x + 4\), we set \(x+4 = 0\), so \(x=-4\). We will use \(-4\) for synthetic division. The coefficients of the dividend \(4x^{3}+10x^{2}-25x - 4\) are \(4\), \(10\), \(-25\), and \(-4\).

Step2: Perform synthetic division

Set up the synthetic division as follows:
Bring down the first coefficient (\(4\)):
\[

$$\begin{array}{r|rrrr} -4 & 4 & 10 & -25 & -4 \\ & & & & \\ \hline & 4 & & & \\ \end{array}$$

\]
Multiply the brought - down number (\(4\)) by \(-4\): \(4\times(-4)=-16\). Add this to the next coefficient (\(10\)): \(10+(-16)=-6\)
\[

$$\begin{array}{r|rrrr} -4 & 4 & 10 & -25 & -4 \\ & & -16 & & \\ \hline & 4 & -6 & & \\ \end{array}$$

\]
Multiply \(-6\) by \(-4\): \(-6\times(-4) = 24\). Add this to the next coefficient (\(-25\)): \(-25 + 24=-1\)
\[

$$\begin{array}{r|rrrr} -4 & 4 & 10 & -25 & -4 \\ & & -16 & 24 & \\ \hline & 4 & -6 & -1 & \\ \end{array}$$

\]
Multiply \(-1\) by \(-4\): \(-1\times(-4)=4\). Add this to the last coefficient (\(-4\)): \(-4 + 4 = 0\)
\[

$$\begin{array}{r|rrrr} -4 & 4 & 10 & -25 & -4 \\ & & -16 & 24 & 4 \\ \hline & 4 & -6 & -1 & 0 \\ \end{array}$$

\]
The numbers at the bottom (\(4\), \(-6\), \(-1\), and \(0\)) represent the coefficients of the quotient polynomial and the remainder. Since the remainder is \(0\), the quotient polynomial is of degree \(2\) (one less than the degree of the dividend). The quotient polynomial is \(4x^{2}-6x - 1\) and the remainder is \(0\). So, \(\frac{4x^{3}+10x^{2}-25x - 4}{x + 4}=4x^{2}-6x - 1\) (with a remainder of \(0\)).

Answer:

\(4x^{2}-6x - 1\)