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Chapter 3: Problem 62

When \(2 x^{2}-7 x+9\) is divided by a polynomial, the quotient is \(2 x-3\) and the remainder is \(3 .\) Find the polynomial.

Short Answer

Expert verified
The polynomial we were looking for is \(x-2\).

Step by step solution

01

Understand the Dividend-Divisor-Quotient-Remainder Relations

It's important to know that the following relation holds for dividing polynomials: the dividend equals the product of divisor and quotient plus the remainder.
02

Setup the Equation

Knowing that our dividend is \(2 x^{2}-7 x+9\), our quotient is \(2x-3\), and remainder is \(3\), we can set up the following equation: \(2 x^{2}-7 x+9 = d(x) \cdot (2x-3) + 3\), where \(d(x)\) is the polynomial we are trying to find.
03

Solve the Equation

To solve this equation for \(d(x)\), rearrange it to find \(d(x)\): \(d(x) = \frac{2 x^{2}-7 x+9 - 3}{2x-3}\). Perform the subtraction in the numerator to get: \(d(x) = \frac{2 x^{2}-7 x+6}{2x-3}\)
04

Simplify the Result

The expression in the numerator can be factorized, which gives us: \(d(x) = \frac{(2x-3) (x-2)}{2x-3}\). As \(2x-3\) appears in both numerator and denominator, they can be cancelled to get the polynomial \(d(x) = x-2\)

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Most popular questions from this chapter

Use a graphing utility to graph \(y=\frac{1}{x}, y=\frac{1}{x^{3}},\) and \(\frac{1}{x^{5}}\) in the same viewing rectangle. For odd values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)

In Exercises \(74-77\), use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-2 x^{3}+6 x^{2}+3 x-1$$

In Exercises \(21-26,\) use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=-5 x^{4}+7 x^{2}-x+9$$

What is meant by the end behavior of a polynomial function?

Show that \(-1\) is a lower bound of \(f(x)=x^{3}-53 x^{2}+\) \(103 x-51 .\) Show that 60 is an upper bound. Use this information and a graphing utility to draw a relatively complete graph of \(f\).
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