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Chapter 2: Problem 67

For each polynomial function given: (a) list each real zero and its multiplicity; (b) determine whether the graph touches or crosses at each \(x\) -intercept; (c) find the \(y\) -intercept and a few points on the graph; (d) determine the end behavior; and (e) sketch the graph. $$f(x)=x^{2}(x-2)^{3}(x+3)^{2}$$

Short Answer

Expert verified
Zeros: 0(2), 2(3), -3(2); Cross/touch: Cross at x=2, touch at x=0, -3; y-int: (0,0); End: rise right, fall left.

Step by step solution

01

Find Real Zeros and Their Multiplicities

To find the real zeros, set the polynomial equation equal to zero: \[f(x) = x^2(x-2)^3(x+3)^2 = 0\]The zeros occur when each factor is equal to zero:1. \(x^2 = 0\) gives \(x = 0\) with multiplicity 2.2. \((x-2)^3 = 0\) gives \(x = 2\) with multiplicity 3.3. \((x+3)^2 = 0\) gives \(x = -3\) with multiplicity 2.
02

Determine Graph Behavior at Each Zero

Examine the behavior of the graph at each zero:- At \(x = 0\) (multiplicity 2), the graph *touches* the \(x\)-axis and turns around.- At \(x = 2\) (multiplicity 3), the graph *crosses* the \(x\)-axis and flattens slightly.- At \(x = -3\) (multiplicity 2), the graph *touches* the \(x\)-axis and turns around.
03

Compute the y-Intercept

The \(y\)-intercept is found by evaluating \(f(0)\):\[f(0) = 0^2 (0-2)^3 (0+3)^2 = 0\]Thus, the \(y\)-intercept is \((0,0)\).
04

Find Additional Points

Evaluate the function at a couple of additional points to assist with sketching:- \(f(1) = 1^2 (1-2)^3 (1+3)^2 = 16\)- \(f(-1) = (-1)^2 (-1-2)^3 (-1+3)^2 = -36\)So, two additional points are \((1, 16)\) and \((-1, -36)\).
05

Determine End Behavior

The end behavior of the polynomial is determined by the degree of the function.The degree is \(2 + 3 + 2 = 7\), which is odd, and the leading coefficient (when expanded) will be positive. Therefore:- As \(x \to \infty\), \(f(x) \to \infty\).- As \(x \to -\infty\), \(f(x) \to -\infty\).
06

Sketch the Graph

Using the information from previous steps:1. Plot the x-intercepts \(x = 0, 2, -3\) and note how the graph behaves at these points.2. Include the y-intercept at \((0,0)\).3. Plot the additional points \((1, 16)\) and \((-1, -36)\).4. Sketch the curve making sure the end behavior matches the determined behavior: rising to the right and falling to the left.

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Key Concepts

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

Real Zeros
Real zeros of a polynomial function are the values of \(x\) that make the function equal to zero. These zeros are crucial because they indicate where the graph of the polynomial intersects or touches the x-axis.
In the case of the polynomial \(f(x) = x^2(x-2)^3(x+3)^2\), the real zeros can be determined by setting each factor equal to zero:
  • \(x^2 = 0\): This gives \(x = 0\) with a multiplicity of 2.
  • \((x-2)^3 = 0\): This gives \(x = 2\) with a multiplicity of 3.
  • \((x+3)^2 = 0\): This gives \(x = -3\) with a multiplicity of 2.
Each zero is an x-intercept, and understanding them helps in graphing the function.
Multiplicity
Multiplicity refers to the number of times a particular zero appears in a polynomial's factored form. It reveals important information about the behavior of the graph at each zero.
For \(f(x) = x^2(x-2)^3(x+3)^2\):
  • Zero \(x = 0\) has multiplicity 2, meaning the graph touches the x-axis and turns back at this point.
  • Zero \(x = 2\) has multiplicity 3, indicating the graph crosses the x-axis and flattens out as it passes through.
  • Zero \(x = -3\) also has multiplicity 2, where the graph touches and turns around.
Knowing the multiplicity helps in predicting the graph's movement at each zero.
End Behavior
End behavior describes how the values of a polynomial function behave as \(x\) approaches positive or negative infinity.
The degree and leading coefficient are critical here. The polynomial \(f(x) = x^2(x-2)^3(x+3)^2\) is of degree 7 (sum of multiplicities), which is odd, and its leading coefficient is positive:
  • As \(x \to \infty\), \(f(x) \to \infty\).
  • As \(x \to -\infty\), \(f(x) \to -\infty\).
This information helps shape the overall direction and curvature of the polynomial's graph.
x-intercept
The x-intercept of a graph is the point where the graph crosses or touches the x-axis, meaning the y-value is zero at this point.
For \(f(x) = x^2(x-2)^3(x+3)^2\), the x-intercepts correspond to its real zeros:
  • At \(x = 0\), the graph touches the axis and turns back due to a multiplicity of 2.
  • At \(x = 2\), the graph crosses the axis and flattens because of a multiplicity of 3.
  • At \(x = -3\), the graph again touches and turns back because of a multiplicity of 2.
Understanding x-intercepts and their behavior helps in sketching the accurate path of the graph.
y-intercept
The y-intercept is the point where the graph intersects the y-axis. This occurs when \(x = 0\).
For the polynomial \(f(x) = x^2(x-2)^3(x+3)^2\), the y-intercept is calculated as follows:
  • Substitute \(x = 0\) into the function: \(f(0) = 0^2(0-2)^3(0+3)^2 = 0\).
Thus, the graph intersects the y-axis at the origin, \((0, 0)\).
This point gives a starting reference on the y-coordinate and ensures the graph is plotted accurately.

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