Math  /  Algebra

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Use the long division method to find the result when 4x48x3+x2+7x114 x^{4}-8 x^{3}+x^{2}+7 x-11 is divided by 2x2x52 x^{2}-x-5. If there is a remainder, express the result in the form q(x)+r(x)b(x)q(x)+\frac{r(x)}{b(x)}.
Answer Attempt 1 out of 2 \square Submit Answer

Studdy Solution

STEP 1

1. We are given a polynomial 4x48x3+x2+7x11 4x^4 - 8x^3 + x^2 + 7x - 11 .
2. We need to divide this polynomial by 2x2x5 2x^2 - x - 5 .
3. We will use the long division method.
4. If there is a remainder, express the result in the form q(x)+r(x)b(x) q(x) + \frac{r(x)}{b(x)} .

STEP 2

1. Set up the long division.
2. Divide the leading term of the dividend by the leading term of the divisor.
3. Multiply the entire divisor by the result from step 2 and subtract from the dividend.
4. Repeat steps 2 and 3 until the degree of the remainder is less than the degree of the divisor.
5. Write the final result including any remainder.

STEP 3

Set up the long division by writing 4x48x3+x2+7x11 4x^4 - 8x^3 + x^2 + 7x - 11 under the division symbol and 2x2x5 2x^2 - x - 5 outside the division symbol.

STEP 4

Divide the leading term of the dividend 4x4 4x^4 by the leading term of the divisor 2x2 2x^2 :
4x42x2=2x2 \frac{4x^4}{2x^2} = 2x^2

STEP 5

Multiply the entire divisor 2x2x5 2x^2 - x - 5 by 2x2 2x^2 :
(2x2)(2x2x5)=4x42x310x2 (2x^2)(2x^2 - x - 5) = 4x^4 - 2x^3 - 10x^2
Subtract this result from the original dividend:
(4x48x3+x2+7x11)(4x42x310x2) (4x^4 - 8x^3 + x^2 + 7x - 11) - (4x^4 - 2x^3 - 10x^2)
Simplify:
8x3+x2+7x11(2x310x2)=6x3+11x2+7x11 -8x^3 + x^2 + 7x - 11 - (-2x^3 - 10x^2) = -6x^3 + 11x^2 + 7x - 11

STEP 6

Repeat the division process with the new polynomial 6x3+11x2+7x11-6x^3 + 11x^2 + 7x - 11.
Divide the leading term 6x3-6x^3 by 2x22x^2:
6x32x2=3x \frac{-6x^3}{2x^2} = -3x

STEP 7

Multiply the entire divisor 2x2x5 2x^2 - x - 5 by 3x-3x:
(3x)(2x2x5)=6x3+3x2+15x (-3x)(2x^2 - x - 5) = -6x^3 + 3x^2 + 15x
Subtract this from the current polynomial:
(6x3+11x2+7x11)(6x3+3x2+15x) (-6x^3 + 11x^2 + 7x - 11) - (-6x^3 + 3x^2 + 15x)
Simplify:
11x2+7x11(3x2+15x)=8x28x11 11x^2 + 7x - 11 - (3x^2 + 15x) = 8x^2 - 8x - 11

STEP 8

Repeat the division process with the new polynomial 8x28x118x^2 - 8x - 11.
Divide the leading term 8x28x^2 by 2x22x^2:
8x22x2=4 \frac{8x^2}{2x^2} = 4

STEP 9

Multiply the entire divisor 2x2x5 2x^2 - x - 5 by 44:
(4)(2x2x5)=8x24x20 (4)(2x^2 - x - 5) = 8x^2 - 4x - 20
Subtract this from the current polynomial:
(8x28x11)(8x24x20) (8x^2 - 8x - 11) - (8x^2 - 4x - 20)
Simplify:
8x11+4x+20=4x+9 -8x - 11 + 4x + 20 = -4x + 9

STEP 10

Since the degree of the remainder 4x+9-4x + 9 is less than the degree of the divisor 2x2x52x^2 - x - 5, we stop here.
The quotient is 2x23x+42x^2 - 3x + 4 and the remainder is 4x+9-4x + 9.
The final result is:
2x23x+4+4x+92x2x5 2x^2 - 3x + 4 + \frac{-4x + 9}{2x^2 - x - 5}

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