Question

1. $(r^4 + 4r^3 - 26r^2 - 25r + 70) div (r + 7)$
2. $(5a^3 + 29a^2 + 18a - 15) div (a + 5)$
3. $(n^4 + 2n^3 - 2n - 5) div (n + 2)$

Explanation:

For Problem 6: $(r^4 + 4r^3 - 26r^2 - 25r + 70) \div (r + 7)$

Step1: Set up polynomial long division

Divide leading terms: $\frac{r^4}{r} = r^3$
Multiply divisor: $r^3(r+7)=r^4+7r^3$
Subtract from dividend:
$(r^4 + 4r^3 - 26r^2 - 25r + 70) - (r^4+7r^3) = -3r^3 -26r^2 -25r +70$

Step2: Divide new leading term

$\frac{-3r^3}{r} = -3r^2$
Multiply divisor: $-3r^2(r+7)=-3r^3-21r^2$
Subtract:
$(-3r^3 -26r^2 -25r +70) - (-3r^3-21r^2) = -5r^2 -25r +70$

Step3: Divide next leading term

$\frac{-5r^2}{r} = -5r$
Multiply divisor: $-5r(r+7)=-5r^2-35r$
Subtract:
$(-5r^2 -25r +70) - (-5r^2-35r) = 10r +70$

Step4: Divide final leading term

$\frac{10r}{r} = 10$
Multiply divisor: $10(r+7)=10r+70$
Subtract:
$(10r +70) - (10r+70) = 0$

Step1: Divide leading terms

$\frac{5a^3}{a} = 5a^2$
Multiply divisor: $5a^2(a+5)=5a^3+25a^2$
Subtract:
$(5a^3 + 29a^2 + 18a - 15) - (5a^3+25a^2) = 4a^2 +18a -15$

Step2: Divide new leading term

$\frac{4a^2}{a} = 4a$
Multiply divisor: $4a(a+5)=4a^2+20a$
Subtract:
$(4a^2 +18a -15) - (4a^2+20a) = -2a -15$

Step3: Divide next leading term

$\frac{-2a}{a} = -2$
Multiply divisor: $-2(a+5)=-2a-10$
Subtract:
$(-2a -15) - (-2a-10) = -5$

Step1: Divide leading terms

$\frac{n^4}{n} = n^3$
Multiply divisor: $n^3(n+2)=n^4+2n^3$
Subtract:
$(n^4 + 2n^3 - 2n - 5) - (n^4+2n^3) = 0n^3+0n^2 -2n -5$

Step2: Handle zero coefficients

$\frac{0n^3}{n} = 0n^2$, multiply/divide gives 0, subtract to keep $0n^2 -2n -5$
$\frac{0n^2}{n} = 0n$, multiply/divide gives 0, subtract to keep $-2n -5$

Step3: Divide final leading term

$\frac{-2n}{n} = -2$
Multiply divisor: $-2(n+2)=-2n-4$
Subtract:
$(-2n -5) - (-2n-4) = -1$

Answer:

$r^3 - 3r^2 - 5r + 10$

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For Problem 8: $(5a^3 + 29a^2 + 18a - 15) \div (a + 5)$