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Chapter 2: Problem 42

Use synthetic division to divide $$f(x)=x^{3}-2 x^{2}-x+2 \text { by } x+1$$ Use the result to find all zeros of \(f\)

Short Answer

Expert verified
After performing synthetic division, the simplified polynomial becomes \(x^{2} - 3x + 3 = 0\). The solutions or zeroes of this polynomial, obtained using the quadratic formula, are \(x = 1.5 + 0.866i\) and \(x = 1.5 - 0.866i\).

Step by step solution

01

Set up the synthetic division

Arrange the coefficients and constant of \(f(x)\) in a horizontal line. Write the root of the divisor \(x + 1\), which is \(-1\), on the left side. The set up should look like this: \[ \begin{array} { r | l } -1 & 1 & -2 & -1 & 2 \end{array} \]
02

Perform the synthetic division

By following the synthetic division process: 1. Bring down the first coefficient (1).2. Multiply with \(-1\), which is \(-1\).3. Add it to the next coefficient (-2). The result is -3.4. Repeat the process until you reach the end: \[ \begin{array} { r | l } -1 & 1 & -2 & -1 & 2 \ & -1 & 3 & 4 & -4 \ \hline & 1 & -3 & 3 & -2 \end{array} \]
03

Write out the simplified polynomial

The simplified polynomial can now be written in the form \(x^{2} - 3x + 3\). The remainder \(-2\) indicates that \(x + 1\) is not a factor of \(f(x)\).
04

Find the zeroes

The zeroes of \(f\) are the values of \(x\) such that \(f(x) = 0\). So, set the simplified polynomial equal to zero and solve for \(x\): \(x^{2} - 3x + 3 = 0\). This is a quadratic equation, which can be solved using the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where in this case, \(a = 1\), \(b = -3\), and \(c = 3\). The solutions yield \(x = 1.5 \pm 0.866i\).

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Most popular questions from this chapter

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.

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

Solve each inequality using a graphing utility. $$x^{2}+3 x-10>0$$

What is a rational function?

Find the horizontal asymptote, if there is one, of the graph of rational function. $$h(x)=\frac{12 x^{3}}{3 x^{2}+1}$$
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