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Chapter 11: Problem 37

Graph each polynomial function. Factor first if the expression is not in factored form. Use the rational zeros theorem as necessary. \(f(x)=x^{2}(x-2)(x+3)^{2}\)

Short Answer

Expert verified
Zeros at \( x = 0 \), \( x = 2 \), and \( x = -3 \). The graph touches at \( x = 0 \) and \( x = -3 \), crosses at \( x = 2 \).

Step by step solution

01

Identify the Polynomial

The given polynomial is already in its factored form: \[ f(x) = x^{2}(x-2)(x+3)^{2} \] There is no need for further factoring.
02

Find the Zeros

Set each factor equal to zero and solve for x to find the zeros of the polynomial: \[ x^2 = 0 \rightarrow x = 0 \] \[ x - 2 = 0 \rightarrow x = 2 \] \[ (x + 3)^2 = 0 \rightarrow x = -3 \] Therefore, the zeros are \( x = 0, 2, -3 \).
03

Determine the Multiplicity of Each Zero

Examine the factors to determine the multiplicity of each zero: - The zero \( x = 0 \) has multiplicity 2 because \( x^2 \) appears in the polynomial. - The zero \( x = 2 \) has multiplicity 1 because \( x - 2 \) appears linearly. - The zero \( x = -3 \) has multiplicity 2 because \( (x + 3)^2 \) appears in the polynomial.
04

Analyze the Behavior at Each Zero

Zeros with even multiplicity (\( x = 0 \) and \( x = -3 \)) will touch and bounce off the x-axis. Zeros with odd multiplicity (\( x = 2 \)) will cross the x-axis.
05

Identify the End Behavior

Since the leading term of the polynomial when expanded is \( x^{5} \), the polynomial has odd degree, and the leading coefficient is positive. Therefore: - As \( x \rightarrow -\infty \), \( f(x) \rightarrow -\infty \). - As \( x \rightarrow +\infty \), \( f(x) \rightarrow +\infty \).
06

Sketch the Graph

Place the zeros \( 0, 2, -3 \) on the x-axis. From the end behavior, the graph will fall to the left and rise to the right. At \( x = 0 \), the graph will just touch the x-axis and bounce back up. At \( x = -3 \), the graph will touch the x-axis and bounce back down. At \( x = 2 \), the graph will cross the x-axis.

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Key Concepts

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

Factoring Polynomials
When solving a polynomial function, the first step is identifying if it is in its factored form. A polynomial is in factored form when it is expressed as the product of its factors. For example, the given polynomial is f(x) = x^2(x-2)(x+3)^2. Each part, like x^2, (x-2), and (x+3)^2, is a factor.
Factoring helps in several ways:
  • It simplifies the process of finding zeros.
  • It reveals the polynomial's structure, making graphing easier.
In the exercise above, we already had the polynomial in factored form. If it wasn't, we would have applied various techniques, such as factoring by grouping or using the Rational Zeros Theorem, to break it down.
Finding Zeros
The zeros of a polynomial are the values of x where the polynomial equals zero. They are crucial for graphing because they indicate where the graph intersects the x-axis. To find them:
  • Set each factor in the polynomial equal to zero.
  • Solve the equations.
    In the exercise, we have:
  • From x^2 = 0, we get x = 0.
  • From x - 2 = 0, we get x = 2.
  • From (x + 3)^2 = 0, we get x = -3.
Thus, the polynomial has zeros at x = 0, 2, -3. These points are where the function cuts through or touches the x-axis.
Multiplicity of Zeros
Multiplicity refers to the number of times a particular zero appears due to repeated factors. It influences the graph's behavior at these points:
  • Even multiplicity: The graph touches the x-axis but bounces off instead of crossing it.
  • Odd multiplicity: The graph crosses the x-axis.
In our exercise:
  • Zero x = 0 has a multiplicity of 2, due to the factor x^2.
  • Zero x = 2 has a multiplicity of 1, because of the linear factor x - 2.
  • Zero x = -3 has a multiplicity of 2, due to the factor (x + 3)^2.
So, for x = 0 and x = -3, the graph will touch and bounce off the x-axis. For x = 2, it will cross the x-axis.
End Behavior
End behavior describes how the polynomial function behaves as x approaches infinity or negative infinity. It's mainly determined by the polynomial's leading term. When expanded, the polynomial f(x) = x^2(x-2)(x+3)^2 becomes a fifth-degree polynomial (highest power is 5) with a positive leading coefficient.
Thus, its end behavior is:
  • As x 鈫� -鈭�, f(x) 鈫� -鈭�.
  • As x 鈫� +鈭�, f(x) 鈫� +鈭�.
This means the graph will fall on the left and rise on the right. Understanding the end behavior helps set the general shape of the graph before plotting the finer details like zeros and their multiplicities.

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Most popular questions from this chapter

Show that the real zeros of each polynomial function satisfy the given conditions. \(f(x)=3 x^{4}+2 x^{3}-4 x^{2}+x-1 ;\) no real zero less than -2

Fill in the blanks with the correct responses: By the definition of an even function, if \((a, b)\) lies on the graph of an even function, then so does \((-a, b)\). Therefore, the graph of an even function is symmetric with respect to the ____. If \((a, b)\) lies on the graph of an odd function, then by definition, so does \((-a,-b)\). Therefore, the graph of an odd function is symmetric with respect to the ____.

Graph each rational function. $$f(x)=\frac{1}{(x+5)(x-2)}$$

Approximate to the nearest hundredth the coordinates of the turning point in the given interval of the graph of each polynomial function. \(f(x)=x^{3}+4 x^{2}-8 x-8, \quad[-3.8,-3]\)

Consider the following "monster" rational function. $$f(x)=\frac{x^{4}-3 x^{3}-21 x^{2}+43 x+60}{x^{4}-6 x^{3}+x^{2}+24 x-20}$$ Analyzing this function will synthesize many of the concepts of this and earlier sections. (a) What is the common factor in the numerator and the denominator? (b) For what value of \(x\) will there be a point of discontinuity (a hole)?
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