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Chapter 2: Problem 81

Simplify the rational expression by using long division or synthetic division. \(\frac{4 x^{3}-8 x^{2}+x+3}{2 x-3}\)

Short Answer

Expert verified
The simplified rational expression is \(2x^2 - x + 1\).

Step by step solution

01

Setting up the Long Division

First, set up a long division. Place \(4x^3 -8x^2 + x +3\) inside the division symbol and \(2x-3\) on the outside.
02

Performing Long Division

Begin dividing each term in the numerator by the first term of the denominator, and proceed step-by-step until you have divided each term in the numerator by the first term in the denominator.
03

Divide \(4x^3\) by \(2x\)

The result is \(2x^2\). Write this result above the division symbol, aligned with the term in the numerator having the same degree. Subtract \(4x^3 - (2x^2)*(2x)\). This equals zero.
04

Continue Division

Next, bring down the next term of the numerator, which is \(-8x^2\). Now we subtract \(-8x^2 - (2x^2)*(-3)\). This equals \(-2x^2\). Dividing \(-2x^2\) by \(2x\) equals \(-x\). Write \(-x\) in the quotient next to \(2x^2\).
05

Finish the Long Division

Next, bring down the next term of the numerator, which is \(x\). Subtract \(x - (-x)*(2x)\) equals \(3x\). Now, continue this process by bringing down 3. Subtract \(3-(-x)*(-3)\) equals zero. Thus our quotient is \(2x^2 - x + 1\) and the remainder is zero.

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