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Chapter 3: Problem 34

Determine the end behavior of the graph of the function. \(n(x)=-2(x+4)(3 x-1)^{3}(x+5)\)

Short Answer

Expert verified
As \(x \to +finity\), \(n(x) \to -finity\). As \(x \to -finity\), \(n(x) \to +finity\).

Step by step solution

01

Identify the Leading Term

Expand the given function to identify the leading term. The given function is: \[n(x) = -2(x+4)(3x-1)^3(x+5)\].We need to consider the highest degree terms from each factor. The leading term will come from multiplying these highest degree terms.
02

Multiply the Highest Degree Terms

The highest degree terms from each factor in the function are:\((x+4) \to x \)\((3x-1)^3 \to (3x)^3 = 27x^3\)\((x+5) \to x \).Now, multiply these terms together along with the constant -2:\[-2 \times x \times 27x^3 \times x = -2 \times 27x^5 = -54x^5\].So, the leading term is \[-54x^5\].
03

Determine End Behavior

To find the end behavior, analyze the leading term \(-54x^5\). Since the exponent is odd and the coefficient is negative:For \(x \to +finity\):\[n(x) \to -finity\].For \(x \to -finity\):\[n(x) \to +finity\].

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Key Concepts

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

Leading Term Identification
In analyzing polynomial functions, one of the first steps is to identify the leading term. This term is critical as it dictates the end behavior of the function. The leading term of a polynomial is the term with the highest degree.

For the given polynomial function: '

This function is given in factored form. To find the leading term, take the highest degree term from each factor.

  • (x+4) contributes x.
  • (3x-1)^3 contributes (3x)^3 = 27x^3.
  • (x+5) contributes x.


Multiply these terms together along with the constant -2, we get:



Therefore, the leading term is -54x^5. This leading term is crucial for determining the polynomial's end behavior.
Expansion of Polynomial Functions
The process of expanding a polynomial function involves expressing it in standard form by multiplying out all its factors. This step is essential for identifying the leading term which provides insights into the function鈥檚 overall nature.

In the provided exercise, the polynomial function is: To expand it, we start by considering the highest degree terms from each factor:

  • (x+4) contributes x.
  • (3x-1)^3 contributes (3x)^3 = 27x^3.
  • (x+5) contributes x.

Multiply these terms together to form the highest degree part of the polynomial:


Notice the importance of including the constant -2 in the multiplication:

This expanded term is the leading term necessary for further analysis.
End Behavior Analysis
Understanding the end behavior of polynomial functions helps in graphing and predicting how the function behaves as x approaches positive or negative infinity. This behavior is dominated by the function鈥檚 leading term.

For the function in question, the leading term is:
This term is a fifth-degree polynomial with a leading coefficient of -54.

Since the degree (5) is odd and the leading coefficient (-54) is negative:

  • As x approaches positive infinity , the term -54x^5 becomes larger but negative, making the function approach -.
  • As x approaches negative infinity , the term -54x^5 also becomes large but positive (since raising a negative number to an odd degree results in a negative value, and multiplying by a negative makes it positive), making the function approach .

This analysis reveals the polynomial鈥檚 end behavior. For large positive values of x, the function decreases without bound (). For large negative values of x, the function increases without bound ().

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Most popular questions from this chapter

Determine if the statement is true or false. If 5 is an upper bound for the real zeros of \(f(x)\), then 4 is also an upper bound.

Let \(a, b,\) and \(c\) represent positive real numbers, where \(a0\) c. Solve \(g(x)<0\).

Explain how the solution set to the inequality \(f(x)<0\) is related to the graph of \(y=f(x)\).

a. Factor the polynomial over the set of real numbers. b. Factor the polynomial over the set of complex numbers. $$f(x)=x^{4}-6 x^{3}+9 x^{2}-6 x+8$$

The procedure to solve a polynomial or rational inequality may be applied to all inequalities of the form \(f(x)>0, f(x)<0,\) \(f(x) \geq 0,\) and \(f(x) \leq 0 .\) That is, find the real solutions to the related equation and determine restricted values of \(x .\) Then determine the sign of \(f(x)\) on each interval defined by the boundary points. Use this process to solve the inequalities. $$ \sqrt{5-x}-7 \geq 0 $$
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