Question

use synthetic division to divide the two polynomials.
(dfrac{5y^5 + 42y^4 - 27y^3 - 2y^2 - 23y - 70}{y + 9})
is the divisor given in ((x - r)) form? yes
how many terms are in the dividend? 6
enter the quotient and remainder.
quotient (\boxed{} +) remainder (\boxed{-25})
synthetic division table
-9
5
42
-27
-2
-23
-70
5
-45
27
0
18
45
5
-3
0
-2
-5
-25

Explanation:

Step1: Identify r for divisor

Divisor is $y+9 = y-(-9)$, so $r=-9$.

Step2: List dividend coefficients

Dividend: $5y^5 + 42y^4 -27y^3 -2y^2 -23y -70$, coefficients are $5, 42, -27, -2, -23, -70$.

Step3: Perform synthetic division

Bring down 5. Multiply by -9: $5\times(-9)=-45$. Add to next coefficient: $42+(-45)=-3$.
Multiply -3 by -9: $(-3)\times(-9)=27$. Add to next coefficient: $-27+27=0$.
Multiply 0 by -9: $0\times(-9)=0$. Add to next coefficient: $-2+0=-2$.
Multiply -2 by -9: $(-2)\times(-9)=18$. Add to next coefficient: $-23+18=-5$.
Multiply -5 by -9: $(-5)\times(-9)=45$. Add to last coefficient: $-70+45=-25$.

Step4: Write quotient polynomial

The resulting coefficients (excluding remainder) are $5, -3, 0, -2, -5$, so the quotient is a 4th-degree polynomial: $5y^4 -3y^3 +0y^2 -2y -5 = 5y^4 -3y^3 -2y -5$.

Answer:

Quotient: $5y^4 - 3y^3 - 2y - 5$
Remainder: $-25$