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Chapter 3: Problem 39

Graph each polynomial function. Factor first if the expression is not in factored form. $$f(x)=x^{3}-x^{2}-2 x$$

Short Answer

Expert verified
Factor: \( f(x) = x(x - 2)(x + 1) \). Roots: 0, 2, -1. Graph passes through these roots.

Step by step solution

01

Identify the Polynomial

First, identify the given polynomial function. The function is given as: \[ f(x) = x^3 - x^2 - 2x \]
02

Factor the Polynomial

Factor the expression to simplify the function for graphing. \[ f(x) = x^3 - x^2 - 2x \] Start by factoring out the greatest common factor (GCF), which is \( x \): \[ f(x) = x(x^2 - x - 2) \]
03

Factor the Quadratic

Next, factor the quadratic expression \( x^2 - x - 2 \). Look for two numbers that multiply to -2 and add to -1. These numbers are -2 and 1: \[ x^2 - x - 2 = (x - 2)(x + 1) \]
04

Write the Factored Form

Write the completely factored form of the polynomial: \[ f(x) = x(x - 2)(x + 1) \]
05

Determine the Roots

Find the roots of the polynomial by setting each factor equal to zero: \[ x = 0, \ x - 2 = 0 \Rightarrow x = 2, \ x + 1 = 0 \Rightarrow x = -1 \] So, the roots are: 0, 2, and -1.
06

Determine the End Behavior

Analyze the end behavior of the polynomial, given that it is a cubic function. As \( x \to \infty \), \( f(x) \to \infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \).
07

Plot Key Points and Roots

Plot the roots on the x-axis: 0, 2, and -1. Also, find and plot additional points by choosing values of \( x \) and finding corresponding \( y \) values.
08

Draw the Graph

Using the plotted points and the end behavior, draw the curve of the polynomial. It should start from \( -\infty \), pass through the points \( (-1, 0) \), \( (0, 0) \), and \( (2, 0) \), and continue to \( \infty \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Polynomials
To start graphing a polynomial function, factoring is a vital first step. In the given exercise, we have the polynomial function
B y factoring out the greatest common factor (GCF), in this case, x, we simplify:
Polynomial Roots
Roots, also known as zeros, are the x-values where the polynomial equals zero. From our factored form, setting each factor to zero, we get the roots: Second Factor: $$x(x - 2)(x + 1) = 0 .$$ ( x detail: = 0, x
First Factor Texted:
Text<\br>F Form and<\br> = 2, x + 1 = 0 => x = -1). = formula < /nSo, the roots are 0, 2, and -1.
End Behavior of Polynomials
The end behavior of a polynomial describes what happens to the value of the function as x approaches very large positive or negative values (i.e., as x approaches infinity or negative infinity). This is crucial for sketching the overall shape of the graph effectively. In our cubic polynomial, as ex=approaches ,f x=f x^3+x^2-x2x-x2x=> y=Inff x^3+.

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Most popular questions from this chapter

Work each problem.What happens to \(y\) if \(y\) varies directly as \(x,\) and \(x\) is halved?

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Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary. $$f(x)=x^{4}-8 x^{3}+29 x^{2}-66 x+72$$

See Exercise \(117 .\) Suppose an economist determines that $$R(x)=\frac{60 x-6000}{x-120}$$ where \(y=R(x)\) is government revenue in tens of millions of dollars for a tax rate of \(x\) percent, with \(y=R(x)\) valid for \(50 \leq x \leq 100 .\) Find the revenue for each tax rate. (a) \(50 \%\) (b) \(60 \%\) (c) \(80 \%\) (d) \(100 \%\) (e) Graph \(R\) in the window \([0,100]\) by \([0,50]\)
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