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Chapter 2: Problem 83

Simplify the rational expression by using long division or synthetic division. \(\frac{x^{4}+6 x^{3}+11 x^{2}+6 x}{x^{2}+3 x+2}\)

Short Answer

Expert verified
After proper simplification using long division, the rational expression simplifies to \(x^{2}+3x+2\), which is the quotient obtained in this case.

Step by step solution

01

Setup for Long Division

Organize the numerator and the denominator in the form of long division, where the numerator (dividend) is \(x^{4}+6 x^{3}+11 x^{2}+6 x\) and the denominator(divisor) is \(x^{2}+3 x+2\).
02

Perform Long Division

Divide the first term of the dividend (\(x^{4}\)) by the first term of the divisor (\(x^{2}\)) to get the first term of the quotient( \(x^{2}\)). Multiply the divisor by this quotient and subtract the result from the dividend. Bring down the next term from the dividend. Repeat this process until all terms from the dividend have been brought down and subtracted off.
03

Verify Your Result

The final quotient is the simplified form of the original expression. To verify the answer, multiply the divisor and the quotient, if the result is equal to the original dividend then the solution is correct.
04

Remove Common Factors

Check if there are any common factors between the quotient and divisor that could be simplified even further. If there are, divide both terms by the common factor to reach the simplest form of the expression.

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Most popular questions from this chapter

Find the domain of \(x\) in the expression. Use a graphing utility to verify your result. \(\sqrt{\frac{x}{x^{2}-9}}\)

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) \(y \geq 36\)

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Solve the inequality and graph the solution on the real number line. \(x^{2}-3 x-18>0\)

Use a graphing utility to graph the equation. Use the graph to approximate the values of \(x\) that satisfy each inequality. \(y=\frac{1}{8} x^{3}-\frac{1}{2} x \quad\) (a) \(y \geq 0 \quad\) (b) \(y \leq 6\)
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