Question

A polynomial function g(x) has a positive leading coefficient. Certain values of g(x) are given in the following table. x –4 –1 0 1 5 8 12 g(x) 0 3 1 2 0 –3 0 If every x-intercept of g(x) is shown in the table and each has a multiplicity of one, what is the end behavior of g(x)?

Answer 1

Using the Factor Theorem and limits, the end behavior of g(x) is that the function decreases to the left and increases to the right.

What is the Factor Theorem?

The Factor Theorem states that a polynomial function with roots $x_1, x_2, \codts, x_n$ is given by:

$f(x) = a(x - x_1)(x - x_2) \cdots (x - x_n)$

In which a is the leading coefficient.

Considering the table, the roots are given as follows:

$x_1 = -4, x_2 = 5, x_3 = 12$

Hence the function is:

f(x) = a(x + 4)(x - 5)(x - 12).

f(x) = a(x² - x - 20)(x - 12)

f(x) = a(x³ - 13x² - 32x + 240).

When x = 0, y = 1, hence the leading coefficient is found as follows:

240a = 1

a = 0.004167

Then:

f(x) = 0.004167(x³ - 13x² - 32x + 240).

The end behavior is given by the limits of f(x) as x goes to infinity, hence:

• $\lim_{x \rightarrow -\infty} f(x) = \lim_{x \rightarrow -\infty} 0.004167 x^3 = -\infty$.

• $\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} 0.004167 x^3 = \infty$.

Hence the end behavior is that the function decreases to the left and increases to the right.

More can be learned about the Factor Theorem at brainly.com/question/24380382

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