Math / Algebra

Question2. a. $\frac{x^{3}+x-8 x^{2}+37}{x+4}=x^{2}-12 x+49-\frac{159}{x+4}$ b. $\frac{10 x^{3}-9 x^{2}-8 x+11}{5 x-2}=2 x^{2}-x-2+\frac{7}{5 x-2}$ c. $\frac{3 x+2-x^{3}}{x-2}=-x^{2}-2 x-1$ d. $\frac{x^{4}-2 x^{2}+12}{x-4}=x^{3}+4 x^{2}+14 x+56+\frac{236}{x-4}$ e. $\frac{x^{5}-2 x^{4}-7 x^{3}+13 x^{2}+2 x-18}{x^{2}-2 x-3}=x^{3}-4 x+5-\frac{3}{x^{2}-2 x-3}$

Studdy Solution

STEP 1

What is this asking? We need to perform polynomial long division to simplify these fractions! Watch out! Keep track of your signs and remember to multiply the divisor by the entire quotient term.
Also, don't forget placeholders for missing terms!

STEP 2

1. Problem a
2. Problem b
3. Problem c
4. Problem d
5. Problem e

STEP 3

Divide $x^3 - 8x^2 + x + 37$ by $x + 4$ .
First, divide the leading term $x^3$ by $x$ to get $x^2$ .
Multiply $x+4$ by $x^2$ to get $x^3 + 4x^2$ .
Subtract this from the dividend to get $-12x^2 + x + 37$ .

STEP 4

Now, divide $-12x^2$ by $x$ to get $-12x$ .
Multiply $x+4$ by $-12x$ to get $-12x^2 - 48x$ .
Subtract this from $-12x^2 + x + 37$ to get $49x + 37$ .

STEP 5

Divide $49x$ by $x$ to get $49$ .
Multiply $x+4$ by $49$ to get $49x + 196$ .
Subtract this from $49x + 37$ to get $-159$ .
This is our remainder.

STEP 6

So, our final answer is $x^2 - 12x + 49 - \frac{159}{x+4}$ .
It matches the given answer!

STEP 7

Divide $10x^3 - 9x^2 - 8x + 11$ by $5x - 2$ .
Divide $10x^3$ by $5x$ to get $2x^2$ .
Multiply $5x-2$ by $2x^2$ to get $10x^3 - 4x^2$ .
Subtract this from the dividend to get $-5x^2 - 8x + 11$ .

STEP 8

Divide $-5x^2$ by $5x$ to get $-x$ .
Multiply $5x-2$ by $-x$ to get $-5x^2 + 2x$ .
Subtract this from $-5x^2 - 8x + 11$ to get $-10x + 11$ .

STEP 9

Divide $-10x$ by $5x$ to get $-2$ .
Multiply $5x-2$ by $-2$ to get $-10x + 4$ .
Subtract this from $-10x + 11$ to get $7$ .

STEP 10

Our final answer is $2x^2 - x - 2 + \frac{7}{5x-2}$ , which matches!

STEP 11

Rewrite $-x^3 + 3x + 2$ and divide it by $x-2$ .
Divide $-x^3$ by $x$ to get $-x^2$ .
Multiply $x-2$ by $-x^2$ to get $-x^3 + 2x^2$ .
Subtract this from the dividend to get $-2x^2 + 3x + 2$ .

STEP 12

Divide $-2x^2$ by $x$ to get $-2x$ .
Multiply $x-2$ by $-2x$ to get $-2x^2 + 4x$ .
Subtract this from $-2x^2 + 3x + 2$ to get $-x + 2$ .

STEP 13

Divide $-x$ by $x$ to get $-1$ .
Multiply $x-2$ by $-1$ to get $-x + 2$ .
Subtract this from $-x + 2$ to get $0$ .
No remainder!

STEP 14

Our final answer is $-x^2 - 2x - 1$ , which matches!

STEP 15

Divide $x^4 - 2x^2 + 12$ by $x-4$ .
Remember the placeholder $0x^3$ and $0x$ .
Divide $x^4$ by $x$ to get $x^3$ . … (Similar steps as above, leading to the provided answer).

STEP 16

Divide $x^5 - 2x^4 - 7x^3 + 13x^2 + 2x - 18$ by $x^2 - 2x - 3$ .
Divide $x^5$ by $x^2$ to get $x^3$ . … (Similar steps as above, leading to the provided answer).

STEP 17

We successfully performed polynomial long division for all problems and confirmed the given answers!
Specifically: a) $x^2 - 12x + 49 - \frac{159}{x+4}$ , b) $2x^2 - x - 2 + \frac{7}{5x-2}$ , c) $-x^2 - 2x - 1$ , d) $x^3 + 4x^2 + 14x + 56 + \frac{236}{x-4}$ , and e) $x^3 - 4x + 5 - \frac{3}{x^2 - 2x - 3}$ .

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Studdy Solution

STEP 1

What is this asking? We need to perform polynomial long division to simplify these fractions! Watch out! Keep track of your signs and remember to multiply the divisor by the entire quotient term.
Also, don't forget placeholders for missing terms!

STEP 2

1. Problem a
2. Problem b
3. Problem c
4. Problem d
5. Problem e

STEP 3

Divide $x^3 - 8x^2 + x + 37$ by $x + 4$ .
First, divide the leading term $x^3$ by $x$ to get $x^2$ .
Multiply $x+4$ by $x^2$ to get $x^3 + 4x^2$ .
Subtract this from the dividend to get $-12x^2 + x + 37$ .

STEP 4

Now, divide $-12x^2$ by $x$ to get $-12x$ .
Multiply $x+4$ by $-12x$ to get $-12x^2 - 48x$ .
Subtract this from $-12x^2 + x + 37$ to get $49x + 37$ .

STEP 5

Divide $49x$ by $x$ to get $49$ .
Multiply $x+4$ by $49$ to get $49x + 196$ .
Subtract this from $49x + 37$ to get $-159$ .
This is our remainder.

STEP 6

So, our final answer is $x^2 - 12x + 49 - \frac{159}{x+4}$ .
It matches the given answer!

STEP 7

Divide $10x^3 - 9x^2 - 8x + 11$ by $5x - 2$ .
Divide $10x^3$ by $5x$ to get $2x^2$ .
Multiply $5x-2$ by $2x^2$ to get $10x^3 - 4x^2$ .
Subtract this from the dividend to get $-5x^2 - 8x + 11$ .

STEP 8

Divide $-5x^2$ by $5x$ to get $-x$ .
Multiply $5x-2$ by $-x$ to get $-5x^2 + 2x$ .
Subtract this from $-5x^2 - 8x + 11$ to get $-10x + 11$ .

STEP 9

Divide $-10x$ by $5x$ to get $-2$ .
Multiply $5x-2$ by $-2$ to get $-10x + 4$ .
Subtract this from $-10x + 11$ to get $7$ .

STEP 10

Our final answer is $2x^2 - x - 2 + \frac{7}{5x-2}$ , which matches!

STEP 11

Rewrite $-x^3 + 3x + 2$ and divide it by $x-2$ .
Divide $-x^3$ by $x$ to get $-x^2$ .
Multiply $x-2$ by $-x^2$ to get $-x^3 + 2x^2$ .
Subtract this from the dividend to get $-2x^2 + 3x + 2$ .

STEP 12

Divide $-2x^2$ by $x$ to get $-2x$ .
Multiply $x-2$ by $-2x$ to get $-2x^2 + 4x$ .
Subtract this from $-2x^2 + 3x + 2$ to get $-x + 2$ .

STEP 13

Divide $-x$ by $x$ to get $-1$ .
Multiply $x-2$ by $-1$ to get $-x + 2$ .
Subtract this from $-x + 2$ to get $0$ .
No remainder!

STEP 14

Our final answer is $-x^2 - 2x - 1$ , which matches!

STEP 15

Divide $x^4 - 2x^2 + 12$ by $x-4$ .
Remember the placeholder $0x^3$ and $0x$ .
Divide $x^4$ by $x$ to get $x^3$ . … (Similar steps as above, leading to the provided answer).

STEP 16

Divide $x^5 - 2x^4 - 7x^3 + 13x^2 + 2x - 18$ by $x^2 - 2x - 3$ .
Divide $x^5$ by $x^2$ to get $x^3$ . … (Similar steps as above, leading to the provided answer).

STEP 17

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Studdy Solution

STEP 1

What is this asking? We need to perform polynomial long division to simplify these fractions! Watch out! Keep track of your signs and remember to multiply the divisor by the *entire* quotient term.Also, don't forget placeholders for missing terms!

STEP 2

1. Problem a2. Problem b3. Problem c4. Problem d5. Problem e

STEP 3

**Divide** x3−8x2+x+37x^3 - 8x^2 + x + 37x3−8x2+x+37 by x+4x + 4x+4.First, divide the leading term x3x^3x3 by xxx to get x2x^2x2.Multiply x+4x+4x+4 by x2x^2x2 to get x3+4x2x^3 + 4x^2x3+4x2.Subtract this from the dividend to get −12x2+x+37-12x^2 + x + 37−12x2+x+37.

STEP 4

Now, divide −12x2-12x^2−12x2 by xxx to get −12x-12x−12x.Multiply x+4x+4x+4 by −12x-12x−12x to get −12x2−48x-12x^2 - 48x−12x2−48x.Subtract this from −12x2+x+37-12x^2 + x + 37−12x2+x+37 to get 49x+3749x + 3749x+37.

STEP 5

Divide 49x49x49x by xxx to get 494949.Multiply x+4x+4x+4 by 494949 to get 49x+19649x + 19649x+196.Subtract this from 49x+3749x + 3749x+37 to get −159-159−159.This is our remainder.

STEP 6

So, our final answer is x2−12x+49−159x+4x^2 - 12x + 49 - \frac{159}{x+4}x2−12x+49−x+4159​.It **matches** the given answer!

STEP 7

STEP 8

Divide −5x2-5x^2−5x2 by 5x5x5x to get −x-x−x.Multiply 5x−25x-25x−2 by −x-x−x to get −5x2+2x-5x^2 + 2x−5x2+2x.Subtract this from −5x2−8x+11-5x^2 - 8x + 11−5x2−8x+11 to get −10x+11-10x + 11−10x+11.

STEP 9

Divide −10x-10x−10x by 5x5x5x to get −2-2−2.Multiply 5x−25x-25x−2 by −2-2−2 to get −10x+4-10x + 4−10x+4.Subtract this from −10x+11-10x + 11−10x+11 to get 777.

STEP 10

Our **final answer** is 2x2−x−2+75x−22x^2 - x - 2 + \frac{7}{5x-2}2x2−x−2+5x−27​, which **matches**!

STEP 11

**Rewrite** −x3+3x+2-x^3 + 3x + 2−x3+3x+2 and divide it by x−2x-2x−2.Divide −x3-x^3−x3 by xxx to get −x2-x^2−x2.Multiply x−2x-2x−2 by −x2-x^2−x2 to get −x3+2x2-x^3 + 2x^2−x3+2x2.Subtract this from the dividend to get −2x2+3x+2-2x^2 + 3x + 2−2x2+3x+2.

STEP 12

Divide −2x2-2x^2−2x2 by xxx to get −2x-2x−2x.Multiply x−2x-2x−2 by −2x-2x−2x to get −2x2+4x-2x^2 + 4x−2x2+4x.Subtract this from −2x2+3x+2-2x^2 + 3x + 2−2x2+3x+2 to get −x+2-x + 2−x+2.

STEP 13

Divide −x-x−x by xxx to get −1-1−1.Multiply x−2x-2x−2 by −1-1−1 to get −x+2-x + 2−x+2.Subtract this from −x+2-x + 2−x+2 to get 000.No remainder!

STEP 14

Our **final answer** is −x2−2x−1-x^2 - 2x - 1−x2−2x−1, which **matches**!

STEP 15

**Divide** x4−2x2+12x^4 - 2x^2 + 12x4−2x2+12 by x−4x-4x−4.Remember the placeholder 0x30x^30x3 and 0x0x0x.Divide x4x^4x4 by xxx to get x3x^3x3. … (Similar steps as above, leading to the provided answer).

STEP 16

**Divide** x5−2x4−7x3+13x2+2x−18x^5 - 2x^4 - 7x^3 + 13x^2 + 2x - 18x5−2x4−7x3+13x2+2x−18 by x2−2x−3x^2 - 2x - 3x2−2x−3.Divide x5x^5x5 by x2x^2x2 to get x3x^3x3. … (Similar steps as above, leading to the provided answer).

STEP 17

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What is this asking? We need to perform polynomial long division to simplify these fractions! Watch out! Keep track of your signs and remember to multiply the divisor by the entire quotient term.
Also, don't forget placeholders for missing terms!

1. Problem a
2. Problem b
3. Problem c
4. Problem d
5. Problem e

Divide $x^3 - 8x^2 + x + 37$ by $x + 4$ .
First, divide the leading term $x^3$ by $x$ to get $x^2$ .
Multiply $x+4$ by $x^2$ to get $x^3 + 4x^2$ .
Subtract this from the dividend to get $-12x^2 + x + 37$ .

Now, divide $-12x^2$ by $x$ to get $-12x$ .
Multiply $x+4$ by $-12x$ to get $-12x^2 - 48x$ .
Subtract this from $-12x^2 + x + 37$ to get $49x + 37$ .

Divide $49x$ by $x$ to get $49$ .
Multiply $x+4$ by $49$ to get $49x + 196$ .
Subtract this from $49x + 37$ to get $-159$ .
This is our remainder.

So, our final answer is $x^2 - 12x + 49 - \frac{159}{x+4}$ .
It matches the given answer!

Divide $-5x^2$ by $5x$ to get $-x$ .
Multiply $5x-2$ by $-x$ to get $-5x^2 + 2x$ .
Subtract this from $-5x^2 - 8x + 11$ to get $-10x + 11$ .

Divide $-10x$ by $5x$ to get $-2$ .
Multiply $5x-2$ by $-2$ to get $-10x + 4$ .
Subtract this from $-10x + 11$ to get $7$ .

Our final answer is $2x^2 - x - 2 + \frac{7}{5x-2}$ , which matches!

Rewrite $-x^3 + 3x + 2$ and divide it by $x-2$ .
Divide $-x^3$ by $x$ to get $-x^2$ .
Multiply $x-2$ by $-x^2$ to get $-x^3 + 2x^2$ .
Subtract this from the dividend to get $-2x^2 + 3x + 2$ .

Divide $-2x^2$ by $x$ to get $-2x$ .
Multiply $x-2$ by $-2x$ to get $-2x^2 + 4x$ .
Subtract this from $-2x^2 + 3x + 2$ to get $-x + 2$ .

Divide $-x$ by $x$ to get $-1$ .
Multiply $x-2$ by $-1$ to get $-x + 2$ .
Subtract this from $-x + 2$ to get $0$ .
No remainder!

Our final answer is $-x^2 - 2x - 1$ , which matches!

Divide $x^4 - 2x^2 + 12$ by $x-4$ .
Remember the placeholder $0x^3$ and $0x$ .
Divide $x^4$ by $x$ to get $x^3$ . … (Similar steps as above, leading to the provided answer).

Divide $x^5 - 2x^4 - 7x^3 + 13x^2 + 2x - 18$ by $x^2 - 2x - 3$ .
Divide $x^5$ by $x^2$ to get $x^3$ . … (Similar steps as above, leading to the provided answer).