Certainly! Let's find the simplest polynomial function with the given roots [tex]\(-3\)[/tex], [tex]\(2\)[/tex], and [tex]\(4\)[/tex].
1. Identify the Roots:
The simplest polynomial with the given roots [tex]\(-3\)[/tex], [tex]\(2\)[/tex], and [tex]\(4\)[/tex] can be expressed using the factor form of a polynomial, which is [tex]\(P(x) = (x - \text{root1})(x - \text{root2})(x - \text{root3})\)[/tex].
2. Express the Polynomial with Factors:
Substituting the given roots, we get:
[tex]\[
P(x) = (x + 3)(x - 2)(x - 4)
\][/tex]
3. Expand the Polynomial:
To find the polynomial [tex]\(P(x)\)[/tex] in standard form, we need to expand the expression [tex]\((x + 3)(x - 2)(x - 4)\)[/tex].
4. Step-by-Step Expansion:
- First, expand [tex]\((x + 3)(x - 2)\)[/tex]:
[tex]\[
(x + 3)(x - 2) = x^2 - 2x + 3x - 6 = x^2 + x - 6
\][/tex]
- Now, multiply this result by [tex]\((x - 4)\)[/tex]:
[tex]\[
(x^2 + x - 6)(x - 4)
\][/tex]
- Distribute each term of [tex]\(x^2 + x - 6\)[/tex] by [tex]\(x - 4\)[/tex]:
[tex]\[
x^2(x - 4) + x(x - 4) - 6(x - 4)
\][/tex]
- Calculating each term:
[tex]\[
x^2 \cdot x = x^3
\][/tex]
[tex]\[
x^2 \cdot (-4) = -4x^2
\][/tex]
[tex]\[
x \cdot x = x^2
\][/tex]
[tex]\[
x \cdot (-4) = -4x
\][/tex]
[tex]\[
-6 \cdot x = -6x
\][/tex]
[tex]\[
-6 \cdot (-4) = 24
\][/tex]
- Combine all the terms:
[tex]\[
x^3 - 4x^2 + x^2 - 4x - 6x + 24
\][/tex]
5. Combine Like Terms:
[tex]\[
x^3 - 4x^2 + x^2 - 10x + 24
\][/tex]
Simplify the above expression to obtain the final polynomial:
[tex]\[
x^3 - 3x^2 - 10x + 24
\][/tex]
Therefore, the simplest polynomial function with the given roots [tex]\(-3\)[/tex], [tex]\(2\)[/tex], and [tex]\(4\)[/tex] is:
[tex]\[
P(x) = x^3 - 3x^2 - 10x + 24
\][/tex]
