To find a polynomial function of degree 3 with the given numbers as zeros [tex]\(-3\)[/tex], [tex]\(6\)[/tex], and [tex]\(5\)[/tex] and assuming the leading coefficient is [tex]\(1\)[/tex], we can follow these steps:
1. Identify the Zeros: The polynomial has zeros at [tex]\(-3\)[/tex], [tex]\(6\)[/tex], and [tex]\(5\)[/tex].
2. Write the Polynomial in Factored Form:
Since the polynomial has the given zeros, it can be represented as:
[tex]\[
P(x) = (x + 3)(x - 6)(x - 5)
\][/tex]
3. Expand the Factored Form:
We need to expand the factored form to get the polynomial in standard form [tex]\(P(x)\)[/tex].
First, expand [tex]\((x + 3)(x - 6)\)[/tex]:
[tex]\[
(x + 3)(x - 6) = x^2 - 6x + 3x - 18 = x^2 - 3x - 18
\][/tex]
Next, expand the result with the remaining factor [tex]\((x - 5)\)[/tex]:
[tex]\[
P(x) = (x^2 - 3x - 18)(x - 5)
\][/tex]
Expand this product step-by-step:
[tex]\[
P(x) = x^2(x - 5) + (-3x)(x - 5) + (-18)(x - 5)
\][/tex]
Simplify each term:
[tex]\[
P(x) = x^3 - 5x^2 - 3x^2 + 15x - 18x + 90
\][/tex]
Combine like terms:
[tex]\[
P(x) = x^3 - 8x^2 - 3x + 90
\][/tex]
So, the polynomial function is:
[tex]\[
P(x) = x^3 - 8x^2 - 3x + 90
\][/tex]
Thus, the polynomial function [tex]\(P(x)\)[/tex] with the given zeros [tex]\(-3\)[/tex], [tex]\(6\)[/tex], and [tex]\(5\)[/tex] is:
[tex]\[
P(x) = x^3 - 8x^2 - 3x + 90
\][/tex]
