Question

use synthetic division to find the quotient and remainder when $-x^4 + 3x^3 + 10x^2 - 4$ is divided by $x - 5$ by completing the parts below.
(a) complete this synthetic division table.
$\begin{array}{r|rrrrr}5) & -1 & 3 & 10 & 0 & -4 \\\\ & & square & square & square & square \\\\ hline & square & square & square & square & square \\\\ end{array}$
(b) write your answer in the following form: quotient$+ \frac{\text{remainder}}{x - 5}$.
$\frac{-x^4 + 3x^3 + 10x^2 - 4}{x - 5} = \square + \frac{\square}{x - 5}$

Explanation:

Part (a)

Step1: Bring down the leading coefficient

Bring down the -1 (the coefficient of \(x^4\)) to the bottom row.
Bottom row: \(-1\)

Step2: Multiply by the divisor root (5) and add

Multiply -1 by 5: \(-1\times5 = -5\). Add to the next coefficient (3): \(3 + (-5)= -2\).
Bottom row: \(-1, -2\)

Step3: Repeat multiplication and addition

Multiply -2 by 5: \(-2\times5 = -10\). Add to the next coefficient (10): \(10 + (-10)= 0\).
Bottom row: \(-1, -2, 0\)

Step4: Repeat multiplication and addition

Multiply 0 by 5: \(0\times5 = 0\). Add to the next coefficient (0): \(0 + 0 = 0\).
Bottom row: \(-1, -2, 0, 0\)

Step5: Repeat multiplication and addition

Multiply 0 by 5: \(0\times5 = 0\). Add to the last coefficient (-4): \(-4 + 0 = -4\).
Bottom row: \(-1, -2, 0, 0, -4\)

The synthetic division table filled is:

$$
LATEXBLOCK0
$$

Part (b)

The coefficients of the quotient (from the bottom row, excluding the last number which is the remainder) correspond to a polynomial of degree one less than the dividend. The dividend is degree 4, so the quotient is degree 3.

The coefficients are -1 (for \(x^3\)), -2 (for \(x^2\)), 0 (for \(x\)), and 0 (constant term). So the quotient is \(-x^3 - 2x^2 + 0x + 0=-x^3 - 2x^2\). The remainder is -4.

Answer:

(a) The synthetic division table is:

$$
LATEXBLOCK0
$$

(b) \(\frac{-x^4 + 3x^3 + 10x^2 - 4}{x - 5}=\boldsymbol{-x^3 - 2x^2}+\frac{\boldsymbol{-4}}{x - 5}\)