Question

algebra 2
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dividing polynomials
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divide.

1. $(r^{3}+2r^{2}-7r - 12)div(r + 3)$
2. $(n^{3}+5n^{2}+10n + 6)div(n + 1)$
3. $(x^{4}+16x^{3}+67x^{2}+63x - 70)div(x + 10)$
4. $(v^{4}+9v^{3}+8v^{2}-17v - 10)div(v + 2)$
5. $(x^{4}-5x^{3}+10x^{2}-3x - 2)div(x - 1)$
6. $(k^{4}-9k^{3}+24k^{2}-20k + 11)div(k - 4)$

Explanation:

Step1: Use polynomial long - division for the first problem

We divide $r^{3}+2r^{2}-7r - 12$ by $r + 3$.
First, divide the leading term of the dividend $r^{3}$ by the leading term of the divisor $r$ to get $r^{2}$. Multiply $r + 3$ by $r^{2}$ to get $r^{3}+3r^{2}$. Subtract this from the dividend: $(r^{3}+2r^{2}-7r - 12)-(r^{3}+3r^{2})=-r^{2}-7r - 12$.
Then divide the leading term of the new dividend $-r^{2}$ by $r$ to get $-r$. Multiply $r + 3$ by $-r$ to get $-r^{2}-3r$. Subtract this from the new dividend: $(-r^{2}-7r - 12)-(-r^{2}-3r)=-4r - 12$.
Finally, divide the leading term of the new dividend $-4r$ by $r$ to get $-4$. Multiply $r + 3$ by $-4$ to get $-4r-12$. Subtract this from the new dividend: $(-4r - 12)-(-4r - 12)=0$. So, $(r^{3}+2r^{2}-7r - 12)\div(r + 3)=r^{2}-r - 4$.

Step2: Use polynomial long - division for the second problem

Divide $n^{3}+5n^{2}+10n + 6$ by $n + 1$.
Divide the leading term of the dividend $n^{3}$ by the leading term of the divisor $n$ to get $n^{2}$. Multiply $n + 1$ by $n^{2}$ to get $n^{3}+n^{2}$. Subtract this from the dividend: $(n^{3}+5n^{2}+10n + 6)-(n^{3}+n^{2})=4n^{2}+10n + 6$.
Divide the leading term of the new dividend $4n^{2}$ by $n$ to get $4n$. Multiply $n + 1$ by $4n$ to get $4n^{2}+4n$. Subtract this from the new dividend: $(4n^{2}+10n + 6)-(4n^{2}+4n)=6n + 6$.
Divide the leading term of the new dividend $6n$ by $n$ to get $6$. Multiply $n + 1$ by $6$ to get $6n+6$. Subtract this from the new dividend: $(6n + 6)-(6n + 6)=0$. So, $(n^{3}+5n^{2}+10n + 6)\div(n + 1)=n^{2}+4n + 6$.

Step3: Use polynomial long - division for the third problem

Divide $x^{4}+16x^{3}+67x^{2}+63x - 70$ by $x + 10$.
Divide the leading term of the dividend $x^{4}$ by the leading term of the divisor $x$ to get $x^{3}$. Multiply $x + 10$ by $x^{3}$ to get $x^{4}+10x^{3}$. Subtract this from the dividend: $(x^{4}+16x^{3}+67x^{2}+63x - 70)-(x^{4}+10x^{3})=6x^{3}+67x^{2}+63x - 70$.
Divide the leading term of the new dividend $6x^{3}$ by $x$ to get $6x^{2}$. Multiply $x + 10$ by $6x^{2}$ to get $6x^{3}+60x^{2}$. Subtract this from the new dividend: $(6x^{3}+67x^{2}+63x - 70)-(6x^{3}+60x^{2})=7x^{2}+63x - 70$.
Divide the leading term of the new dividend $7x^{2}$ by $x$ to get $7x$. Multiply $x + 10$ by $7x$ to get $7x^{2}+70x$. Subtract this from the new dividend: $(7x^{2}+63x - 70)-(7x^{2}+70x)=-7x - 70$.
Divide the leading term of the new dividend $-7x$ by $x$ to get $-7$. Multiply $x + 10$ by $-7$ to get $-7x-70$. Subtract this from the new dividend: $(-7x - 70)-(-7x - 70)=0$. So, $(x^{4}+16x^{3}+67x^{2}+63x - 70)\div(x + 10)=x^{3}+6x^{2}+7x - 7$.

Step4: Use polynomial long - division for the fourth problem

Divide $v^{4}+9v^{3}+8v^{2}-17v - 10$ by $v + 2$.
Divide the leading term of the dividend $v^{4}$ by the leading term of the divisor $v$ to get $v^{3}$. Multiply $v + 2$ by $v^{3}$ to get $v^{4}+2v^{3}$. Subtract this from the dividend: $(v^{4}+9v^{3}+8v^{2}-17v - 10)-(v^{4}+2v^{3})=7v^{3}+8v^{2}-17v - 10$.
Divide the leading term of the new dividend $7v^{3}$ by $v$ to get $7v^{2}$. Multiply $v + 2$ by $7v^{2}$ to get $7v^{3}+14v^{2}$. Subtract this from the new dividend: $(7v^{3}+8v^{2}-17v - 10)-(7v^{3}+14v^{2})=-6v^{2}-17v - 10$.
Divide the leading term of the new dividend $-6v^{2}$ by $v$ to get $-6v$. Multiply $v + 2$ by $-6v$ to get $-6v^{2}-12v$. Subtract this from the new dividend: $(-6v^{2}-17v - 10)-(-6v^{2}-12v)=-5v - 10$.
Divide the leading term of the new dividend $-5v$ by $v$ to get $-5$. Multiply $v + 2$ by $-5$ to get $-5v-10$. Subtract this from the new dividend: $(-5v - 10)-(-5v - 10)=0$. So, $(v^{4}+9v^{3}+8v^{2}-17v - 10)\div(v + 2)=v^{3}+7v^{2}-6v - 5$.

Step5: Use polynomial long - division for the fifth problem

Divide $x^{4}-5x^{3}+10x^{2}-3x - 2$ by $x - 1$.
Divide the leading term of the dividend $x^{4}$ by the leading term of the divisor $x$ to get $x^{3}$. Multiply $x - 1$ by $x^{3}$ to get $x^{4}-x^{3}$. Subtract this from the dividend: $(x^{4}-5x^{3}+10x^{2}-3x - 2)-(x^{4}-x^{3})=-4x^{3}+10x^{2}-3x - 2$.
Divide the leading term of the new dividend $-4x^{3}$ by $x$ to get $-4x^{2}$. Multiply $x - 1$ by $-4x^{2}$ to get $-4x^{3}+4x^{2}$. Subtract this from the new dividend: $(-4x^{3}+10x^{2}-3x - 2)-(-4x^{3}+4x^{2})=6x^{2}-3x - 2$.
Divide the leading term of the new dividend $6x^{2}$ by $x$ to get $6x$. Multiply $x - 1$ by $6x$ to get $6x^{2}-6x$. Subtract this from the new dividend: $(6x^{2}-3x - 2)-(6x^{2}-6x)=3x - 2$.
Divide the leading term of the new dividend $3x$ by $x$ to get $3$. Multiply $x - 1$ by $3$ to get $3x-3$. Subtract this from the new dividend: $(3x - 2)-(3x - 3)=1$. So, $(x^{4}-5x^{3}+10x^{2}-3x - 2)\div(x - 1)=x^{3}-4x^{2}+6x + 3+\frac{1}{x - 1}$.

Step6: Use polynomial long - division for the sixth problem

Divide $k^{4}-9k^{3}+24k^{2}-20k + 11$ by $k - 4$.
Divide the leading term of the dividend $k^{4}$ by the leading term of the divisor $k$ to get $k^{3}$. Multiply $k - 4$ by $k^{3}$ to get $k^{4}-4k^{3}$. Subtract this from the dividend: $(k^{4}-9k^{3}+24k^{2}-20k + 11)-(k^{4}-4k^{3})=-5k^{3}+24k^{2}-20k + 11$.
Divide the leading term of the new dividend $-5k^{3}$ by $k$ to get $-5k^{2}$. Multiply $k - 4$ by $-5k^{2}$ to get $-5k^{3}+20k^{2}$. Subtract this from the new dividend: $(-5k^{3}+24k^{2}-20k + 11)-(-5k^{3}+20k^{2})=4k^{2}-20k + 11$.
Divide the leading term of the new dividend $4k^{2}$ by $k$ to get $4k$. Multiply $k - 4$ by $4k$ to get $4k^{2}-16k$. Subtract this from the new dividend: $(4k^{2}-20k + 11)-(4k^{2}-16k)=-4k + 11$.
Divide the leading term of the new dividend $-4k$ by $k$ to get $-4$. Multiply $k - 4$ by $-4$ to get $-4k + 16$. Subtract this from the new dividend: $(-4k + 11)-(-4k + 16)=-5$. So, $(k^{4}-9k^{3}+24k^{2}-20k + 11)\div(k - 4)=k^{3}-5k^{2}+4k - 4-\frac{5}{k - 4}$.

Answer:

1. $r^{2}-r - 4$
2. $n^{2}+4n + 6$
3. $x^{3}+6x^{2}+7x - 7$
4. $v^{3}+7v^{2}-6v - 5$
5. $x^{3}-4x^{2}+6x + 3+\frac{1}{x - 1}$
6. $k^{3}-5k^{2}+4k - 4-\frac{5}{k - 4}$