Question

use synthetic division to find $(5x^3 - 26x^2 + 21) \div (x - 5)$. 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: Identify coefficients and root

For the polynomial \(5x^3 - 26x^2 + 0x + 21\) (we add the \(0x\) term for the missing \(x\) term) and the divisor \(x - 5\), the root is \(x = 5\). The coefficients are \(5\), \(-26\), \(0\), \(21\).

Step2: Set up synthetic division

Bring down the first coefficient:
\[

$$\begin{array}{r|rrrr} 5 & 5 & -26 & 0 & 21 \\ & & 25 & -5 & -25 \\ \hline & 5 & -1 & -5 & -4 \\ \end{array}$$

\]
Multiply the root (5) by the brought - down coefficient (5) to get 25, add to -26: \(-26 + 25=-1\).
Multiply 5 by -1 to get -5, add to 0: \(0+( - 5)=-5\).
Multiply 5 by -5 to get -25, add to 21: \(21+( - 25)=-4\).

Step3: Write the quotient and remainder

The quotient polynomial \(q(x)\) has coefficients \(5\), \(-1\), \(-5\), so \(q(x)=5x^2 - x - 5\). The remainder \(r=-4\) and the divisor \(d(x)=x - 5\).

Answer:

\(5x^{2}-x - 5+\frac{-4}{x - 5}\) (or \(5x^{2}-x - 5-\frac{4}{x - 5}\))