Question

The table gives a partial set of values of a polynomial h(x), which has a leading coefficient of 1. 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

From the table, we were given the zero of the function h(x).

The zeros of the function are

`-3,-2,\text{ and 1}`

The formula to obtain the equation of the function is,

`(x-a)(x-b)(x-c)`

Where

`\begin{gathered} a=-3 \\ b=-2 \\ c=1 \end{gathered}`

Hence,

`\begin{gathered} h(x)=(x--3)(x--2)(x-1) \\ h(x)=(x+3)(x+2)(x-1) \end{gathered}`

Expanding the function above

`\begin{gathered} h(x)=x(x+2)+3(x+2)(x-1) \\ h(x)=x^2+2x+3x+6(x-1) \\ h(x)=x^2+5x+6(x-1) \\ h(x)=x(x^2+5x+6)-1(x^2+5x+6) \\ h(x)=x^3+5x^2+6x-x^2-5x-6 \\ h(x)=x^3+5x^2-x^2+6x-5x-6 \\ h(x)=x^3+4x^2+x-6 \end{gathered}`

Therefore, the equation of the polynomial function is

`h(x)=x^3+4x^2+x-6`

Hence, the answer is Option 3.