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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 of \(\frac{4 x^{3}-8 x^{2}+x+3}{2 x-3}\) is \(2x^{2}+x-1\).

Step by step solution

01

Organize the equation for long division

Arrange the given equation as a division problem \(4x^{3}-8x^{2}+x+3 \div 2x-3\). So 2x-3 will be the divisor and \(4x^{3}-8x^{2}+x+3\) will be the dividend.
02

Perform long division

Divide \(4x^{3}\) by \(2x\) to get \(2x^{2}\), which is the first part of the quotient. Multiply \(2x^{2}\) by \(2x-3\) to give \(4x^{3}-6x^{2}\). Subtract this from \(4x^{3}-8x^{2}\) to obtain \(-2x^{2}\). Bring down the 'x' from the dividend to obtain, \(-2x^{2}+x\).Repeat the process of dividing, multiplying and subtracting. The final quotient will contain \(2x^{2} + x - 1\), and the remainder will be 0.
03

Write the answer

Since there is no remainder after the long division process, the simplified rational expression will be the quotient \(2x^{2}+x-1\).

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