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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
\[x^2 - 15x + 24 - \frac{284}{x^2 - 4}\] is the simplified form of the expression.

Step by step solution

01

Preparation

Write the polynomial long division similar to a regular long division problem. The numerator goes under the long division bar, and the denominator goes to the left of it. It should be set up as follows: \(x^{2}-4 \) into \(x^{4}+9 x^{3}-5 x^{2}-36 x+4 \). Return the highest order coefficient of the denominator (1 in this case because the denominator is \(x^2\)) into the first term of the numerator (\(x^4\)), and divide.
02

Apply Long Division

The first step gives you \(x^2\). To continue with the division, we multiply \(x^2\) by the divisor \(x^2 - 4\) to get \(x^4 - 4x^2\). Subtract this from the polynomial under the division bar, giving \(15x^2-36x+4\). At this point, we bring down \(-36x\) and \('+4'\). Now we repeat the process by dividing the first term of this new polynomial by the first term of the divisor. This gives us \(-15x\).
03

Continue Long Division

Multiply \(-15x\) by \(x^2 - 4\) to get \(-15x^3 + 60x\). Subtract this expression from our current polynomial under the division bar, giving \(24x^3 - 24x + 4\). Again, repeat the division process to get the term \(24x\).
04

Finalize Long Division

Multiply \(24x\) by \(x^2 - 4\) to get \(24x^3 - 96x\). Subtract this expression from our current polynomial under the division bar, giving \(72x + 4\). Lastly, repeat the division process once more to get the term \(72\). At the end we have a remainder of \(-284\). Thus, the simplified expression is \(x^2 - 15x + 24 - \frac{284}{x^2 - 4}\).

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Most popular questions from this chapter

Use a graphing utility to graph the equation. Use the graph to approximate the values of \(x\) that satisfy each inequality. \(y=x^{3}-x^{2}-16 x+16 \quad\) (a) \(y \leq 0 \quad\) (b) \(y \geq 36\)

Determine whether each value of \(x\) is a solution of the inequality. Inequality. \(x^{2}-x-12 \geq 0 \quad\) (a) \(x=5 \quad\) (b) \(x=0\) (c) \(x=-4\) (d) \(x=-3\)

Find the domain of \(x\) in the expression. Use a graphing utility to verify your result. \(\sqrt{\frac{x}{x^{2}-2 x-35}}\)

Solve the inequality. (Round your answers to two decimal places.) \(\frac{1}{2.3 x-5.2}>3.4\)

Solve the inequality and graph the solution on the real number line. \(\frac{4 x-1}{x}>0\)
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