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Chapter 2: Problem 61

State the Remainder Theorem.

Short Answer

Expert verified
The Remainder Theorem states that if a polynomial \(f(x)\) is divided by a linear divisor of the form \(x-a\), then the remainder can be found by evaluating the polynomial at the point \(x = a\), i.e., \(f(a)\). This means that the remainder when a polynomial \(f(x)\) is divided by \(x-a\) is the same as the result of \(f(a)\).

Step by step solution

01

Understanding the Remainder Theorem

The Remainder Theorem is a result in algebra which states that if a polynomial \(f(x)\) is divided by a linear divisor of the form \(x-a\), then the remainder can be found by simply substituting \(a\) into the function \(f\). This means the Remainder is given by \(f(a)\).
02

An Example of the Remainder Theorem

For instance, if you have a polynomial \(f(x) = x^3 + 2x^2 - 3x + 2\) and you're asked to find the remainder when divided by \(x-2\), you just substitute \(2\) into the polynomial i.e., \(f(2)\). Calculating this gives a value of 14, which is our remainder.

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Most popular questions from this chapter

Use a graphing utility to graph \(y=\frac{1}{x^{2}}, y=\frac{1}{x^{4}},\) and \(y=\frac{1}{x^{6}}\) in the same viewing rectangle. For even values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)

Find the horizontal asymptote, if there is one, of the graph of rational function. $$h(x)=\frac{15 x^{3}}{3 x^{2}+1}$$

a. If \(y=k x^{2},\) find the value of \(k\) using \(x=2\) and \(y=64\). b. Substitute the value for \(k\) into \(y=k x^{2}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=5\).

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has a vertical asymptote given by \(x=1,\) a slant asymptote whose equation is \(y=x, y\) -intercept at \(2,\) and \(x\)-intercepts at -1 and 2.

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}-1}{x}$$
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