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Chapter 2: Problem 25

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

Short Answer

Expert verified
The result of the division is \(x\).

Step by step solution

01

Recognize the Polynomials

The first polynomial is \(x^{4}\), the second one is \((x-1)^{3}\). The task is to divide the first one by the second one using long division.
02

Set-up the Long Division

Write the first polynomial, \(x^{4}\), under the long division symbol and the second polynomial, \((x-1)^{3}\), goes on the outside of the symbol. This is just like long division with numbers.
03

Perform the First Division

To get the first term of the quotient, divide the leading term of the dividend (\(x^{4}\)) by the leading term of the divisor ( \(x^{3}\) from the expanded \((x-1)^{3}\)). The result is \(x\).
04

Multiply and Subtract

Multiply the divisor \((x-1)^{3}\) by \(x\) (got from step 3) and subtract the resulting product from the original polynomial \(x^{4}\), then drag down the next term. This process is repeated until all terms have been accounted for, or when the degree of the remaining polynomial is less than the degree of the divisor. Since we started with a polynomial of degree 4, and our divisor is of degree 3, we stop here, because the degree of the remainder is smaller than that of the divisor
05

Write Down the Remainder

Any remaining polynomial left after the final subtraction is the remainder and is written as a fraction over the original divisor. Since in this case the degree of the remainder is 0 and the divisor -1, we don't have a remainder

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Most popular questions from this chapter

Write the polynomial as the product of linear factors and list all the zeros of the function. $$h(x)=x^{4}+6 x^{3}+10 x^{2}+6 x+9$$

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$x^{2}+b x-4=0$$

Use the information in the table to answer each question. $$\begin{array}{|c|c|} \hline \text { Interval } & \text { Value of } f(x) \\\\\hline(-\infty,-2) & \text { Positive } \\\\\hline(-2,1) & \text { Negative } \\\\\hline(1,4) & \text { Negative } \\\\\hline(4, \infty) & \text { Positive } \\\\\hline\end{array}$$ (a) What are the three real zeros of the polynomial function \(f ?\) (b) What can be said about the behavior of the graph of \(f\) at \(x=1 ?\) (c) What is the least possible degree of \(f ?\) Explain. Can the degree of \(f\) ever be odd? Explain. (d) Is the leading coefficient of \(f\) positive or negative? Explain. (e) Sketch a graph of a function that exhibits the behavior described in the table.

Find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) $$2,5+i$$

For each function, identify the degree of the function and whether the degree of the function is even or odd. Identify the leading coefficient and whether the leading coefficient is positive or negative. Use a graphing utility to graph each function. Describe the relationship between the degree of the function and the sign of the leading coefficient of the function and the right-hand and left-hand behavior of the graph of the function. (a) \(f(x)=x^{3}-2 x^{2}-x+1\) (b) \(f(x)=2 x^{5}+2 x^{2}-5 x+1\) (c) \(f(x)=-2 x^{5}-x^{2}+5 x+3\) (d) \(f(x)=-x^{3}+5 x-2\) (e) \(f(x)=2 x^{2}+3 x-4\) (f) \(f(x)=x^{4}-3 x^{2}+2 x-1\) (g) \(f(x)=x^{2}+3 x+2\)
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