Question

use synthetic division to simplify \\(\frac{x^3 + x^2}{x + 1}\\).
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 root of divisor

For \(x + 1\), set \(x+1 = 0\), so \(x=-1\). The coefficients of dividend \(x^{3}+x^{2}+0x + 0\) (adding missing terms) are \(1,1,0,0\).

Step2: Perform synthetic division

Set up synthetic division with root \(-1\) and coefficients \(1,1,0,0\):

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The quotient polynomial \(q(x)\) is \(x^{2}+0x + 0=x^{2}\), the remainder \(r = 0\), and the divisor \(d(x)=x + 1\).

Step3: Write in the required form

Using the form \(q(x)+\frac{r}{d(x)}\), we substitute the values: \(x^{2}+\frac{0}{x + 1}\), which simplifies to \(x^{2}+0=x^{2}\) (but since \(r = 0\), the form is \(x^{2}+\frac{0}{x + 1}\) or just \(x^{2}\) as the fraction term is \(0\)).

Answer:

\(x^{2}+\frac{0}{x + 1}\) (or simply \(x^{2}\) as the remainder is \(0\))