To find a polynomial function of degree 3 with the given numbers as its zeros, we need to form a polynomial whose roots are [tex]\(-1\)[/tex], [tex]\(3\)[/tex], and [tex]\(6\)[/tex]. The general form of a polynomial with these roots can be written as:
[tex]\[
f(x) = (x - \text{root}_1)(x - \text{root}_2)(x - \text{root}_3)
\][/tex]
Here, the roots are given as [tex]\(-1\)[/tex], [tex]\(3\)[/tex], and [tex]\(6\)[/tex]. So, the polynomial function [tex]\(f(x)\)[/tex] can be written as:
[tex]\[
f(x) = (x - (-1))(x - 3)(x - 6)
\][/tex]
Simplifying the terms inside the parentheses:
[tex]\[
f(x) = (x + 1)(x - 3)(x - 6)
\][/tex]
Next, we will expand this expression step by step:
First, expand [tex]\((x + 1)(x - 3)\)[/tex]:
[tex]\[
(x + 1)(x - 3) = x(x - 3) + 1(x - 3)
\][/tex]
[tex]\[
= x^2 - 3x + x - 3
\][/tex]
[tex]\[
= x^2 - 2x - 3
\][/tex]
Now we need to multiply this result by [tex]\((x - 6)\)[/tex]:
[tex]\[
(x^2 - 2x - 3)(x - 6)
\][/tex]
Distribute each term in the first polynomial by each term in the second polynomial:
[tex]\[
= x^2(x - 6) - 2x(x - 6) - 3(x - 6)
\][/tex]
[tex]\[
= x^3 - 6x^2 - 2x^2 + 12x - 3x + 18
\][/tex]
Combine like terms:
[tex]\[
= x^3 - 8x^2 + 9x + 18
\][/tex]
Thus, the polynomial function of degree 3 with the zeros [tex]\(-1\)[/tex], [tex]\(3\)[/tex], and [tex]\(6\)[/tex] is:
[tex]\[
f(x) = x^3 - 8x^2 + 9x + 18
\][/tex]
