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Chapter 3: Problem 21

Use long division to divide. $$\frac{2 x^{3}-4 x^{2}-15 x+5}{(x-1)^{2}}$$

Short Answer

Expert verified
The short answer (final result) will be obtained at the end of Step 5 after performing polynomial long division and simplification.

Step by step solution

01

Rewrite the polynomial division as long division

First, write the given exercise in the form we usually use to perform normal arithmetic division, but this time we will perform a polynomial division.
02

Polynomial Division - First Round

Now divide the first term in the numerator \(2x^{3}\) with the first term in the denominator \(x^{2}\) to get \(2x\). Then, multiply \((x-1)^{2}\) by \(2x\), and subtract the result from the initial polynomial in the numerator. The result of this operation will replace the initial polynomial.
03

Polynomial Division - Second Round

Perform the division again, now with the first term of the new polynomial obtained in Step 2, and the first term of the denominator \((x-1)^{2}\). Remember to subtract the result from the new polynomial, and repeat this process until no terms are left to divide.
04

Simplification

After all terms have been divided, simplify the obtained expression to its lowest terms.
05

Final Answer

The final result obtained is the answer to the long division of the polynomials.

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Most popular questions from this chapter

Match the cubic function with the correct number of rational and irrational zeros. (a) Rational zeros: \(0 ; \quad\) Irrational zeros: 1 (b) Rational zeros: \(3 ;\) Irrational zeros: \(\mathbf{0}\) (c) Rational zeros: \(1 ; \quad\) Irrational zeros: 2 (d) Rational zeros: \(1 ;\) Irrational zeros: \(\mathbf{0}\) $$f(x)=x^{3}-2$$

Find the rational zeros of the polynomial function. $$f(z)=z^{3}+\frac{11}{6} z^{2}-\frac{1}{2} z-\frac{1}{3}=\frac{1}{6}\left(6 z^{3}+11 z^{2}-3 z-2\right)$$

Use a graphing utility to compare the graphs of \(y_{1}\) and \(y_{2}.\) $$y_{1}=\frac{3 x^{3}-5 x^{2}+4 x-5}{2 x^{2}-6 x+7}, \quad y_{2}=\frac{3 x^{3}}{2 x^{2}}$$ Start with a viewing window of \(-5 \leq x \leq 5\) and \(-10 \leq y \leq 10,\) and then zoom out. Make a conjecture about how the graph of a rational function \(f\) is related to the graph of \(y=a_{n} x^{n} / b_{m} x^{m},\) where \(a_{n} x^{n}\) is the leading term of the numerator of \(f\) and \(b_{m} x^{m}\) is the leading term of the denominator of \(f.\)

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f .\) Then find all real zeros of the function. \(f(x)=2 x^{3}-3 x^{2}-12 x+8\) Upper bound: \(x=4\) Lower bound: \(x=-3\)

Find all real zeros of the polynomial function. $$f(x)=4 x^{4}-55 x^{2}-45 x+36$$
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