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Chapter 2: Problem 80

Explain how to use the Leading Coefficient Test to determine the end behavior of a polynomial function.

Short Answer

Expert verified
The Leading Coefficient Test allows one to determine the graph's end behavior. The degree (even or odd) and leading coefficient's sign (positive or negative) dictate the behavior. The results can either be \(\uparrow, \uparrow\), \(\downarrow, \downarrow\), \(\downarrow, \uparrow\), or \(\uparrow, \downarrow\), which correspond to the specific combinations respectively.

Step by step solution

01

Identify the Degree and Leading Coefficient

Firstly, identify the degree of the polynomial, which is the highest power of the variable \(x\). Then identify the leading coefficient, which is the coefficient of the highest degree term.
02

Use the Leading Coefficient Test

In the Leading Coefficient Test, look at the degree and the sign of the leading coefficient. If the degree is even and the leading coefficient is positive, the end behavior is \(\uparrow , \uparrow\). If the leading coefficient is negative, it is \(\downarrow , \downarrow\). If the degree is odd and the leading coefficient is positive, the end behavior is \(\downarrow , \uparrow\). If it's negative, then it's \(\uparrow , \downarrow\). The first arrow represents as x approaches negative infinity, while the second arrow represents as x approaches positive infinity.
03

Apply to Specific Polynomial

Finally, apply this knowledge to the specific polynomial at hand. Example: for \(f(x) = -2x^3 + x^2 - x + 2\), the leading term is -2x^3, so the leading coefficient is -2 (a negative number), and the degree is 3 (an odd number). Thus, the end behavior is \(\uparrow , \downarrow\).

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