Question

use synthetic division to divide the two polynomials.$dfrac{4x^{4}+15x^{3}-27x^{2}-13x-19}{-x-5}$is the divisor given in $(x - r)$ form? nowhat do you need to multiply the divisor by to get a coefficient of 1?how many terms are in the dividend?enter the quotient and remainder.quotient + remaindersynthetic division table

Explanation:

Step1: Adjust divisor to (x-r) form

Multiply divisor $-x-5$ by $-1$:
$(-1)(-x-5) = x+5 = x - (-5)$
Multiply dividend by $-1$ too:
$(-1)(4x^4+15x^3-27x^2-13x-19) = -4x^4-15x^3+27x^2+13x+19$

Step2: Count dividend terms

List terms: $4x^4, 15x^3, -27x^2, -13x, -19$.
Number of terms: $5$

Step3: Set up synthetic division

Use $r=-5$, coefficients of adjusted dividend: $-4, -15, 27, 13, 19$

$$\begin{array}{r|rrrrr} -5 & -4 & -15 & 27 & 13 & 19 \\ & & 20 & -25 & -10 & -15 \\ \hline & -4 & 5 & 2 & 3 & 4 \\ \end{array}$$

Step4: Get quotient and remainder

Coefficients of quotient: $-4, 5, 2, 3$ → $-4x^3+5x^2+2x+3$
Remainder: $4$
Reverse the sign correction (since we multiplied by -1 earlier):
Quotient: $-(-4x^3+5x^2+2x+3) = 4x^3-5x^2-2x-3$
Remainder: $4$ (remainder sign stays, as $\frac{\text{adjusted dividend}}{\text{adjusted divisor}} = \frac{-\text{original dividend}}{-\text{original divisor}} = \frac{\text{original dividend}}{\text{original divisor}}$)

Answer:

What do you need to multiply the divisor by to get a coefficient of 1? $\boldsymbol{-1}$
How many terms are in the dividend? $\boldsymbol{5}$
Quotient: $\boldsymbol{4x^3 - 5x^2 - 2x - 3}$
Remainder: $\boldsymbol{4}$