Question

use synthetic division to simplify $\frac{x^{5}-25x^{3}-2x^{2}+7x + 15}{x - 5}$. 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

The divisor is $x - 5$, so we use $5$ as the divisor in synthetic - division. The coefficients of the dividend $x^{5}-25x^{3}-2x^{2}+7x + 15$ are $1,0,-25,-2,7,15$.

Step2: Bring down the first coefficient

Bring down the first coefficient $1$:

5 |  1  0  -25  -2  7  15
   |     5  25   0 -10 -15
   |_____________________
     1  5   0  -2 -3   0

Step3: Multiply and add

Multiply the number in the divisor ($5$) by the number just brought down ($1$), get $5$, and add it to the next coefficient ($0$) to get $5$. Repeat this process for all coefficients.

Step4: Write the quotient

The quotient $q(x)$ has coefficients $1,5,0,-2,-3$ and the degree of $q(x)$ is one less than the degree of the dividend. So $q(x)=x^{4}+5x^{3}-2x - 3$ and the remainder $r = 0$.

Answer:

$x^{4}+5x^{3}-2x - 3+\frac{0}{x - 5}=x^{4}+5x^{3}-2x - 3$