Question

use synthetic division to find $(x^3 + 3) \div (x - 2)$. 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: Set up synthetic division

For dividing \(x^3 + 3\) by \(x - 2\), we first write the coefficients of the dividend polynomial. The dividend \(x^3+0x^2 + 0x+3\) (we include the missing \(x^2\) and \(x\) terms with coefficient 0) and the root of the divisor \(x - 2\) is \(x = 2\). So we set up synthetic division as:

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Step2: Interpret the result

The numbers in the bottom row are the coefficients of the quotient polynomial and the remainder. The degree of the quotient polynomial is one less than the dividend. So the quotient \(q(x)=x^2 + 2x+4\) and the remainder \(r = 11\), and the divisor \(d(x)=x - 2\). So the division can be written as \(x^2+2x + 4+\frac{11}{x - 2}\)

Answer:

\(x^{2}+2x + 4+\frac{11}{x - 2}\)