Question

use synthetic division to find the result when $x^4 + x^3 - 14x^2 - 10x$ is divided by $x + 4$. if there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.

Explanation:

Step1: Identify root for divisor

For \(x + 4\), the root is \(x=-4\) (since \(x + 4=0\Rightarrow x=-4\)). The coefficients of the dividend \(x^{4}+x^{3}-14x^{2}-10x + 0\) (we add the constant term \(0\) as there's no constant term) are \(1,1,-14,-10,0\).

Step2: Set up synthetic division

Write the root \(-4\) and the coefficients:

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• Bring down the first coefficient \(1\).
• Multiply \(1\times(-4)=-4\), add to next coefficient: \(1+(-4)=-3\).
• Multiply \(-3\times(-4)=12\), add to next coefficient: \(-14 + 12=-2\).
• Multiply \(-2\times(-4)=8\), add to next coefficient: \(-10+8=-2\).
• Multiply \(-2\times(-4)=8\), add to last coefficient: \(0 + 8 = 8\).

Step3: Write quotient and remainder

The coefficients of the quotient \(q(x)\) are \(1,-3,-2,-2\) (degree \(3\)), so \(q(x)=x^{3}-3x^{2}-2x - 2\). The remainder \(r(x)=8\) and divisor \(b(x)=x + 4\).

Answer:

\(x^{3}-3x^{2}-2x - 2+\frac{8}{x + 4}\)