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Math  /  Algebra

QuestionUse long division to divide. SEE EXAMPLE 1
15. x3+5x2−x−5x^{3}+5 x^{2}-x-5 divided by x−1x-1
16. 2x3+9x2+10x+32 x^{3}+9 x^{2}+10 x+3 divided by 2x+12 x+1
17. 3x3−2x2+7x+93 x^{3}-2 x^{2}+7 x+9 divided by x2−3xx^{2}-3 x
18. 2x4−6x2+32 x^{4}-6 x^{2}+3 divided by 2x−62 x-6
Use synthetic division to divide. SEE EXAMPLE
19. x4−25x2+144x^{4}-25 x^{2}+144 divided by x−4x-4
20. x3+6x2+3x−10x^{3}+6 x^{2}+3 x-10 divided by x+5x+5
21. x5+2x4−3x3+x−1x^{5}+2 x^{4}-3 x^{3}+x-1 divided by x+2x+2
22. −x4+7x3+x2−2x−12-x^{4}+7 x^{3}+x^{2}-2 x-12 divided by x−3x-3

Studdy Solution

STEP 1

1. We are given several polynomial division problems.
2. Problems 15-18 require long division of polynomials.
3. Problems 19-22 require synthetic division of polynomials.

STEP 2

1. Perform long division for each polynomial division problem (15-18).
2. Perform synthetic division for each polynomial division problem (19-22).

STEP 3

For problem 15, divide x3+5x2−x−5 x^3 + 5x^2 - x - 5 by x−1 x - 1 using long division.
1. Divide the first term of the dividend x3 x^3 by the first term of the divisor x x to get x2 x^2 .
2. Multiply the entire divisor x−1 x - 1 by x2 x^2 and subtract the result from the dividend.
3. Bring down the next term and repeat the process until all terms are divided.

STEP 4

For problem 16, divide 2x3+9x2+10x+3 2x^3 + 9x^2 + 10x + 3 by 2x+1 2x + 1 using long division.
1. Divide the first term of the dividend 2x3 2x^3 by the first term of the divisor 2x 2x to get x2 x^2 .
2. Multiply the entire divisor 2x+1 2x + 1 by x2 x^2 and subtract the result from the dividend.
3. Bring down the next term and repeat the process until all terms are divided.

STEP 5

For problem 17, divide 3x3−2x2+7x+9 3x^3 - 2x^2 + 7x + 9 by x2−3x x^2 - 3x using long division.
1. Divide the first term of the dividend 3x3 3x^3 by the first term of the divisor x2 x^2 to get 3x 3x .
2. Multiply the entire divisor x2−3x x^2 - 3x by 3x 3x and subtract the result from the dividend.
3. Bring down the next term and repeat the process until all terms are divided.

STEP 6

For problem 18, divide 2x4−6x2+3 2x^4 - 6x^2 + 3 by 2x−6 2x - 6 using long division.
1. Divide the first term of the dividend 2x4 2x^4 by the first term of the divisor 2x 2x to get x3 x^3 .
2. Multiply the entire divisor 2x−6 2x - 6 by x3 x^3 and subtract the result from the dividend.
3. Bring down the next term and repeat the process until all terms are divided.

STEP 7

For problem 19, use synthetic division to divide x4−25x2+144 x^4 - 25x^2 + 144 by x−4 x - 4 .
1. Write down the coefficients of the dividend: 1,0,−25,0,144 1, 0, -25, 0, 144 .
2. Use the root 4 4 for synthetic division.
3. Perform synthetic division by bringing down the first coefficient and multiplying by the root, adding to the next coefficient, and repeating.

STEP 8

For problem 20, use synthetic division to divide x3+6x2+3x−10 x^3 + 6x^2 + 3x - 10 by x+5 x + 5 .
1. Write down the coefficients of the dividend: 1,6,3,−10 1, 6, 3, -10 .
2. Use the root −5-5 for synthetic division.
3. Perform synthetic division by bringing down the first coefficient and multiplying by the root, adding to the next coefficient, and repeating.

STEP 9

For problem 21, use synthetic division to divide x5+2x4−3x3+x−1 x^5 + 2x^4 - 3x^3 + x - 1 by x+2 x + 2 .
1. Write down the coefficients of the dividend: 1,2,−3,0,1,−1 1, 2, -3, 0, 1, -1 .
2. Use the root −2-2 for synthetic division.
3. Perform synthetic division by bringing down the first coefficient and multiplying by the root, adding to the next coefficient, and repeating.

STEP 10

For problem 22, use synthetic division to divide −x4+7x3+x2−2x−12-x^4 + 7x^3 + x^2 - 2x - 12 by x−3 x - 3 .
1. Write down the coefficients of the dividend: −1,7,1,−2,−12-1, 7, 1, -2, -12 .
2. Use the root 33 for synthetic division.
3. Perform synthetic division by bringing down the first coefficient and multiplying by the root, adding to the next coefficient, and repeating.

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