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Chapter 3: Problem 35

Determine the end behavior of the graph of the function. \(p(x)=-2 x^{2}(3-x)(2 x-3)^{3}\)

Short Answer

Expert verified
As \( x \rightarrow \infty \, p(x) \rightarrow \infty \); as \( x \rightarrow -\infty \, p(x) \rightarrow \infty \).

Step by step solution

01

- Identify Leading Term

Identify the term with the highest degree in the polynomial. The degree of a polynomial determines its end behavior. For the function given, expand the polynomial to find the leading term.
02

- Determine the Degree

The degree of the polynomial is found by summing the exponents of the highest power terms in each factor. For \( p(x) = -2 x^{2}(3-x)(2 x-3)^{3} \), the degree is \( 2 + 1 + 3 = 6 \).
03

- Consider the Leading Coefficient

Determine the coefficient of the highest degree term. In this case, multiply the leading coefficients from each term: \( -2 \times (-1) \times (2^3) = -2 \times -1 \times 8 = 16 \). So the leading term is 16x^6.
04

- Analyze End Behavior Based on Leading Term

Since the polynomial has an even degree (6) and the leading coefficient is positive (16), the end behavior on both ends will be the same. As \( x \rightarrow \infty \, p(x) \rightarrow \infty \); and as \( x \rightarrow -\infty \, p(x) \rightarrow \infty \).

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Key Concepts

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

leading term identification
Before analyzing the end behavior of a polynomial, it is crucial to identify the leading term. The leading term is the term with the highest degree, which dominates the behavior of the polynomial for large values of x. In the function given, \( p(x) = -2 x^{2}(3-x)(2 x-3)^{3} \), expanding the polynomial helps reveal the leading term. By focusing on the highest power of each factor, we can easily find it without multiplying everything out. Identifying the leading term is the first key step in determining the polynomial's end behavior.
degree of polynomial
The degree of a polynomial is the sum of the exponents in the leading term. This degree tells us a lot about the polynomial's end behavior. For \( p(x) = -2 x^{2}(3-x)(2 x-3)^{3} \), the highest power in each factor is \( x^{2} \), \( x \), and \( x^{3} \) respectively. To find the overall degree, we add these exponents together: \( 2 + 1 + 3 = 6 \). The degree being 6, which is even, will influence the end behavior significantly. The degree of the polynomial indicates how the function's graph behaves as \( x \) moves towards positive or negative infinity.
leading coefficient
The leading coefficient is the coefficient of the leading term in a polynomial. It plays a major role in determining the direction of the end behavior. For the given function, when expanding \( -2 x^{2} \) , \( (3-x) \), and \( (2 x-3)^{3} \), we find that multiplying the coefficients gives \( -2 \times -1 \times 8 = 16 \). Hence, the leading term is \( 16x^6 \). The sign of this coefficient (positive in this case) will influence whether the end behavior points upward or downward as \( x \rightarrow \pm \infty \).
end behavior analysis
Analyzing the end behavior of a polynomial involves looking at the degree of the polynomial and the sign of the leading coefficient. Since our polynomial has a degree of 6 (even) and a positive leading coefficient (16), its end behavior is straightforward. For even-degree polynomials with positive leading coefficients, the ends of the graph will both point upwards. Thus,

  • As \( x \rightarrow \infty \), \( p(x) \rightarrow \infty \).

  • As \( x \rightarrow -\infty \), \( p(x) \rightarrow \infty \).

This consistent pattern helps us predict how the polynomial behaves at the extremities of the x-axis, making it easier to sketch and understand the graph.

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Most popular questions from this chapter

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