Question

use synthetic division to find ((x^{4}-8x^{2}+16)div(x + 3)).
write your answer in the form (q(x)+\frac{r}{d(x)}), where (q(x)) is a polynomial, (r) is an integer, and (d(x)) is a linear polynomial. simplify any fractions.

Explanation:

Step1: Identify the root of the divisor

The divisor is \(x + 3\), so we set \(x+3 = 0\) and solve for \(x\). We get \(x=-3\).

Step2: Set up the polynomial for synthetic division

The dividend is \(x^{4}-8x^{2}+16\). We need to write it in standard form with all coefficients (including those of missing terms). The polynomial can be written as \(x^{4}+0x^{3}-8x^{2}+0x + 16\). So the coefficients are \(1,0, - 8,0,16\).

Step3: Perform synthetic division

We use the root \(-3\) and the coefficients \(1,0, - 8,0,16\) for synthetic division:

\[

$$\begin{array}{r|rrrrr} -3 & 1 & 0 & -8 & 0 & 16\\ & & -3 & 9 & -3 & 9\\ \hline & 1 & -3 & 1 & -3 & 25\\ \end{array}$$

\]

• The first number \(1\) is brought down.
• Multiply \(1\times(-3)=-3\), add to the next coefficient: \(0 + (-3)=-3\).
• Multiply \(-3\times(-3) = 9\), add to the next coefficient: \(-8+9 = 1\).
• Multiply \(1\times(-3)=-3\), add to the next coefficient: \(0+(-3)=-3\).
• Multiply \(-3\times(-3)=9\), add to the last coefficient: \(16 + 9=25\).

Step4: Write the quotient and remainder

The coefficients of the quotient polynomial \(q(x)\) are \(1,-3,1,-3\), so \(q(x)=x^{3}-3x^{2}+x - 3\). The remainder \(r = 25\) and the divisor \(d(x)=x + 3\).

Answer:

\(x^{3}-3x^{2}+x - 3+\frac{25}{x + 3}\)