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Chapter 2: Problem 84

Simplify the rational expression by using long division or synthetic division. $$\frac{x^{4}+9 x^{3}-5 x^{2}-36 x+4}{x^{2}-4}$$

Short Answer

Expert verified
The simplified rational expression is \(x^{2}+9x+5+ \frac{56x + 24}{x^{2}- 4}\)

Step by step solution

01

Identify the Numerator and Denominator

In the rational expression \(\frac{x^{4}+9 x^{3}-5 x^{2}-36 x+4}{x^{2}-4}\), the numerator is the polynomial \(x^{4}+9 x^{3}-5 x^{2}-36 x+4\) and the denominator is the polynomial \(x^{2}-4\).
02

Perform Long Division

In this step, perform long division by dividing the numerator by the denominator. The quotient is \(x^{2}+9x+5\) with a remainder of \(56x + 24\).
03

Interpret the Result

The quotient and remainder together give the simplified form of the original expression. The quotient represents the polynomial term while the remainder forms part of a rational fraction over the divisor. The simplified rational expression therefore becomes \(x^{2}+9x+5+ \frac{56x + 24}{x^{2}- 4}\)

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