Question

The table gives a partial set of values of a polynomial h(x), which has a leading coefficient of 1. x –2 0 1 2 3 h(x) 0 –12 0 8 0 If every x-intercept of h(x) is shown in the table and has a multiplicity of one, what is the equation of the polynomial function?

Answer 1

Using the Factor Theorem, the equation of h(x) is given as follows:

h(x) = -2(x³ - 2x² - 5x + 6)

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.

Looking at the table, considering the values of x when h(x) = 0, the roots of h(x) are given as follows:

$x_1 = -2, x_2 = 1, x_3 = 3$

Then the rule is:

h(x) = a(x + 2)(x - 1)(x - 3)

h(x) = a(x² + x - 2)(x - 3)

h(x) = a(x³ - 2x² - 5x + 6)

The h-intercept is of -12, as when x = 0, h = -12, hence this is used to find a as follows:

6a = -12

a = -12/6

a = -2.

Hence the function is given by:

h(x) = -2(x³ - 2x² - 5x + 6)

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

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