Question

use synthetic division to find the quotient and remainder when $x^4 - 25$ is divided by $x - 5$.

select one:
a. quotient: $x^3 - 5$, remainder: 0
b. quotient: $x^3 + 5x^2 + 25x + 125$, remainder: 650
c. quotient: $x^3 + 5x^2 - 5x + 25$, remainder: 600
d. quotient: $x^3 + 5x^2 + 25x + 125$, remainder: 600

Explanation:

Step1: Identify coefficients and root

Dividend coefficients: $1, 0, 0, 0, -25$ (for $x^4+0x^3+0x^2+0x-25$); root of $x-5=0$ is $5$.

Step2: Set up synthetic division

$$\begin{array}{r|rrrrr} 5 & 1 & 0 & 0 & 0 & -25 \\ & & 5 & 25 & 125 & 625 \\ \hline & 1 & 5 & 25 & 125 & 600 \\ \end{array}$$

Step3: Read quotient and remainder

First 4 values = quotient coefficients; last value = remainder.
Quotient: $x^3+5x^2+25x+125$, Remainder: $600$

Answer:

D. Quotient: $x^3 + 5x^2 + 25x + 125$, Remainder: 600