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Chapter 3: Problem 22

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. $$P(x)=(x-3)^{2}(x+1)^{2}$$

Short Answer

Expert verified
The polynomial has roots at x = 3 and x = -1, both with even multiplicity, a y-intercept at (0, 9), and the graph rises on both ends.

Step by step solution

01

Identify the Roots

The given polynomial is \( P(x) = (x-3)^{2}(x+1)^{2} \). The roots of the polynomial are the values of \( x \) that make \( P(x) = 0 \). From the factors, we identify the roots as \( x = 3 \) and \( x = -1 \). Each has multiplicity 2.
02

Determine the Y-intercept

To find the y-intercept, substitute \( x = 0 \) into the polynomial: \( P(0) = (0-3)^{2}(0+1)^{2} = 9 \times 1 = 9 \). Thus, the y-intercept is \( (0, 9) \).
03

Analyze the End Behavior

The polynomial \( P(x) = (x-3)^{2}(x+1)^{2} \) has a degree of 4 (since each squared factor contributes two to the degree). As a positive leading-coefficient, even-degree polynomial, the graph will rise to positive infinity on both sides as \( x \to \pm \infty \).
04

Consider Multiplicities at Roots

Both roots \( x = 3 \) and \( x = -1 \) have even multiplicities (2). Therefore, at these points, the graph will touch the x-axis and turn around (rather than crossing it).
05

Sketch the Graph

To sketch the graph, plot the x-intercepts at \( (3, 0) \) and \( (-1, 0) \), and the y-intercept at \( (0, 9) \). Show the function touching and turning at the x-intercepts due to the even multiplicity of roots. Begin the graph from the upper left, touch the x-axis at \( x = -1 \), rise to the y-intercept at \( x = 0 \), descend to touch the x-axis again at \( x = 3 \), and then rise to the upper right.

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Key Concepts

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

Roots of Polynomial
Roots of a polynomial are values of \( x \) for which the polynomial equals zero. Think of them as the points where the graph of the polynomial touches or crosses the x-axis. In the polynomial \( P(x) = (x-3)^{2}(x+1)^{2} \), it's factored into two products. This allows us to find the roots directly by solving each factor for zero:
  • \( x - 3 = 0 \Rightarrow x = 3 \)
  • \( x + 1 = 0 \Rightarrow x = -1 \)
These roots are \( x = 3 \) and \( x = -1 \). Each factor is squared, giving them a multiplicity of 2. This specific multiplicity affects how the graph behaves at these x-values, which we will explore later.
Y-intercept
The y-intercept of a graph is where the graph crosses the y-axis. It is found by setting \( x = 0 \) and solving for \( y \). For \( P(x) = (x-3)^{2}(x+1)^{2} \), substituting \( x = 0 \) gives:\[ P(0) = (0-3)^{2} (0+1)^{2} = 9 \times 1 = 9 \]Thus, the y-intercept is \( (0, 9) \). This point is critical in plotting the polynomial as it ensures that the graph is positioned correctly in relation to the y-axis. Every polynomial has exactly one y-intercept since it is only dependent on the constant term when the polynomial is expressed in standard form.
End Behavior
End behavior describes how the polynomial behaves as \( x \) approaches positive or negative infinity. For \( P(x) = (x-3)^{2}(x+1)^{2} \), the degree of the polynomial is 4. Each squared factor contributes 2 to this degree, making it even. An even-degree polynomial with a positive leading coefficient (like ours) has ends that go to infinity in the same direction:
  • As \( x \to +\infty \), \( P(x) \to +\infty \)
  • As \( x \to -\infty \), \( P(x) \to +\infty \)
The graph rises on both ends, which is an essential clue for sketching an accurate representation of the polynomial. Recognizing this behavior helps predict the general shape of the graph without evaluating it point-by-point.
Multiplicity of Roots
Multiplicity of a root refers to how many times a particular root is repeated in the polynomial equation. For \( P(x) = (x-3)^{2}(x+1)^{2} \), each root, \( x = 3 \) and \( x = -1 \) , is repeated twice, giving them a multiplicity of 2.Multiplicity affects whether a graph crosses or merely touches the x-axis at the root:
  • Roots with even multiplicity (like in our polynomial) cause the graph to touch the x-axis and turn around without crossing it.
  • Roots with odd multiplicity cross the x-axis and do not turn around.
In our polynomial, at \( x = 3 \) and \( x = -1 \), the graph will touch the axis, turn, and rise back toward the direction indicated by the end behavior, illustrating once again the characteristic of even multiplicity roots.

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Most popular questions from this chapter

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the quadratic formula, or other factoring techniques. $$P(x)=x^{5}-7 x^{4}+9 x^{3}+23 x^{2}-50 x+24$$

Graph the rational function \(f\) and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$f(x)=\frac{-x^{4}+2 x^{3}-2 x}{(x-1)^{2}}, g(x)=1-x^{2}$$

Use Descartes' Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. $$P(x)=2 x^{6}+5 x^{4}-x^{3}-5 x-1$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. $$s(x)=\frac{4 x-8}{(x-4)(x+1)}$$

Use Descartes' Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. $$P(x)=x^{8}-x^{5}+x^{4}-x^{3}+x^{2}-x+1$$
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