Question

use synthetic division to find the result when $x^3 - 6x^2 - 26x + 6$ is divided by $x + 3$. if there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.

Explanation:

Step1: Identify the root for divisor

For the divisor \(x + 3\), set \(x+3 = 0\), so \(x=-3\). We will use synthetic division with root \(-3\) and coefficients of the dividend \(x^{3}-6x^{2}-26x + 6\) which are \(1\), \(-6\), \(-26\), \(6\).

Step2: Perform synthetic division

Set up the synthetic division:

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• Bring down the first coefficient \(1\).
• Multiply \(1\) by \(-3\) to get \(-3\), add to \(-6\): \(-6+(-3)=-9\).
• Multiply \(-9\) by \(-3\) to get \(27\), add to \(-26\): \(-26 + 27=1\).
• Multiply \(1\) by \(-3\) to get \(-3\), add to \(6\): \(6+(-3)=3\).

The last number \(3\) is the remainder, and the other numbers \(1\), \(-9\), \(1\) are the coefficients of the quotient polynomial of degree \(2\) (since the dividend is degree \(3\) and divisor is degree \(1\)), so the quotient is \(x^{2}-9x + 1\) and the remainder is \(3\).

Step3: Write the result

Using the division algorithm \(f(x)=d(x)q(x)+r(x)\), so \(\frac{x^{3}-6x^{2}-26x + 6}{x + 3}=x^{2}-9x + 1+\frac{3}{x + 3}\).

Answer:

\(x^{2}-9x + 1+\frac{3}{x + 3}\)