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Chapter 2: Problem 61

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. $$f(x)=2 x^{5}-6 x^{4}-8 x^{3}$$

Short Answer

Expert verified
Zeros at \(x = 0\) (mult. 3), \(x = 4\), \(x = -1\). Graph crosses at all zeros. \(y\)-intercept is 0. For large \(x\), graph rises to \(\infty\) as \(x\) goes to both \(\pm\infty\). Sketch as determined.

Step by step solution

01

Factor Out Common Terms

Inspect the function \(f(x) = 2x^5 - 6x^4 - 8x^3\). We will first factor out the greatest common factor (GCF). Here, the GCF is \(2x^3\). Therefore, we have:\[ f(x) = 2x^3(x^2 - 3x - 4) \]
02

Factor the Quadratic Expression

Next, factor the quadratic \(x^2 - 3x - 4\) into \[ (x-4)(x+1) \]Thus, the expression is:\[ f(x) = 2x^3(x - 4)(x + 1) \]
03

Identify the Real Zeros and Multiplicities

Find the real zeros by setting each factor to zero:- \(x^3 = 0\) gives zero at \(x = 0\) with multiplicity 3.- \(x - 4 = 0\) gives a zero at \(x = 4\) with multiplicity 1.- \(x + 1 = 0\) gives a zero at \(x = -1\) with multiplicity 1.
04

Determine Touch/Cross at Each X-Intercept

For zero with odd multiplicity, the graph will cross the x-axis. For even multiplicities, it touches. Thus:- At \(x = 0\), the graph touches the x-axis as it has an odd multiplicity (crosses).- At \(x = 4\) and \(x = -1\), the graph crosses the x-axis (since both have odd multiplicity).
05

Find the Y-Intercept and a Few Points

Find the y-intercept by evaluating \(f(0)\): \[ f(0) = 2(0)^3((0) - 4)((0) + 1) = 0 \]Some additional points:- \(f(1) = 2(1)^3(1-4)(1+1) = 2(-3)(2) = -12 \)- \(f(-2) = 2(-2)^3(-2-4)(-2+1) = 2(-8)(-6)(-1) = 96 \)
06

Determine the End Behavior

The function represents a polynomial of degree 5 with a positive leading coefficient. As \(x \to \infty, f(x) \to \infty\) and as \(x \to -\infty, f(x) \to -\infty\).
07

Sketch the Graph

Combine all gathered information to sketch: - Start from the left with the graph going downwards (from negative infinity).- Cross through \(x = -1\).- Come back down, cross through \(x = 0\) flatly (since it touches and crosses).- Rise up again and cross at \(x = 4\) upwards towards positive infinity.

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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 occur where the function equals zero. These points correspond to the x-values for which the polynomial evaluates to zero. In this case, the function is given as \( f(x) = 2x^3(x - 4)(x + 1) \). By solving this factored expression, we find the real zeros by setting each factor to zero:

- For \( 2x^3\), the solution is \( x = 0 \).
- For \( (x - 4) \), the solution is \( x = 4 \).
- For \( (x + 1) \), the solution is \( x = -1 \).

These values are called the real zeros of the function, where the graph intersects or touches the x-axis.
Multiplicity
Multiplicity refers to how many times a particular zero is repeated for a given polynomial function. This concept tells us about the behavior of the graph at the zero. In our function:\( f(x) = 2x^3(x - 4)(x + 1) \), each factor contributes a multiplicity to its zero.

- The zero at \( x = 0 \) has a multiplicity of 3 because of the factor \( 2x^3 \). This multiplicity implies that the graph flattens at, but eventually crosses through this point.
- The zeros at \( x = 4 \) and \( x = -1 \) each have a multiplicity of 1. These zeros are where the graph crosses the x-axis, giving a simple linear passage through the x-axis.
End Behavior
End behavior describes how the graph of a polynomial behaves as \( x \) approaches 卤鈭�. The degree and the leading coefficient determine this behavior.

For the polynomial \( f(x) = 2x^5 - 6x^4 - 8x^3 \), which has a degree of 5 (from the highest power of \( x \) in the polynomial), the leading coefficient is positive (2). Hence, as \( x \) approaches infinity, \( f(x) \) also approaches infinity, and as \( x \) approaches negative infinity, \( f(x) \) approaches negative infinity.

Simply put, the graph starts from the bottom left (coming from -鈭� as x moves negatively) and ends at the top right (going towards +鈭� as x increases).
X-Intercepts
The x-intercepts of a graph are the points where it crosses the x-axis. These points can be identified by the real zeros of the polynomial.

For \( f(x) = 2x^3(x - 4)(x + 1) \):
  • The x-intercept at \( x = 0 \) arises due to the zero from the factor \( 2x^3 \). At this point, the graph touches and crosses the axis due to the odd multiplicity.
  • The intercept \( x = 4 \) stems from \( (x - 4) = 0 \), and the graph crosses at this point.
  • Similarly, \( x = -1 \) from \( (x + 1) = 0 \) is another crossing point.
The crossing-through behavior at each x-intercept indicates the nature and interaction of the zeros with the x-axis.
Factoring Polynomials
Factoring polynomials is a key skill in solving polynomial equations and finding their zeros. It involves expressing the polynomial as a product of its factors.

Starting with \( f(x) = 2x^5 - 6x^4 - 8x^3 \), the first step is to factor out the greatest common factor (GCF), which in this case is \( 2x^3 \). Simplifying gives:
  • \( f(x) = 2x^3(x^2 - 3x - 4) \)
Next, further factor the quadratic \( x^2 - 3x - 4 \):
  • \( x^2 - 3x - 4 \) factors into \( (x - 4)(x + 1) \)
This leads to the fully factored form:
\( f(x) = 2x^3(x - 4)(x + 1) \)

Factoring polynomials allows us to identify zeros easily, understand the multiplicity and behavior at those zeros, and analyze the graph's end behavior.

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