Question

use synthetic division to simplify \\(\frac{x^4 - 5x^3 - 15x^2 + 7x + 12}{x + 2}\\). 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 for synthetic division

For dividing by \(x + 2\), the root \(c=-2\) (since \(x+2 = x - (-2)\)). The coefficients of the dividend \(x^4 - 5x^3 - 15x^2 + 7x + 12\) are \(1, -5, -15, 7, 12\).

Step2: Perform synthetic division

Set up the synthetic division:

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The last number is the remainder \(r = -6\), and the quotient polynomial \(q(x)\) has coefficients \(1, -7, -1, 9\), so \(q(x)=x^3 - 7x^2 - x + 9\) and \(d(x)=x + 2\).

Step3: Write in the required form

Using the form \(q(x)+\frac{r}{d(x)}\), we substitute the values: \(x^3 - 7x^2 - x + 9+\frac{-6}{x + 2}\), which simplifies to \(x^3 - 7x^2 - x + 9-\frac{6}{x + 2}\) (or \(x^3 - 7x^2 - x + 9+\frac{-6}{x + 2}\)).

Answer:

\(x^{3}-7x^{2}-x + 9+\frac{-6}{x + 2}\) (or \(x^{3}-7x^{2}-x + 9-\frac{6}{x + 2}\))