To find a polynomial function of degree 3 with given zeros [tex]\(-2, 3, 5\)[/tex] and a leading coefficient of 1, you should follow these steps:
1. Identify the factors: The given zeros correspond to the factors of the polynomial. For each zero [tex]\(c\)[/tex], the corresponding factor of the polynomial is [tex]\((x - c)\)[/tex].
Thus, the factors will be:
[tex]\[
(x + 2), (x - 3), (x - 5)
\][/tex]
2. Form the polynomial: Multiply these factors together to form the polynomial.
[tex]\[
f(x) = (x + 2)(x - 3)(x - 5)
\][/tex]
3. Expand the polynomial: Use the distributive property (also known as the FOIL method for binomials) to expand these factors step-by-step.
First, multiply the first two factors:
[tex]\[
(x + 2)(x - 3) = x \cdot x + x \cdot (-3) + 2 \cdot x + 2 \cdot (-3) = x^2 - 3x + 2x - 6 = x^2 - x - 6
\][/tex]
Now, we have:
[tex]\[
f(x) = (x^2 - x - 6)(x - 5)
\][/tex]
Next, distribute [tex]\((x - 5)\)[/tex] across [tex]\((x^2 - x - 6)\)[/tex]:
[tex]\[
(x^2 - x - 6)(x - 5) = x^2(x - 5) - x(x - 5) - 6(x - 5)
\][/tex]
Expanding each term:
[tex]\[
x^2(x - 5) = x^3 - 5x^2
\][/tex]
[tex]\[
-x(x - 5) = -x^2 + 5x
\][/tex]
[tex]\[
-6(x - 5) = -6x + 30
\][/tex]
Combine all these terms together:
[tex]\[
f(x) = x^3 - 5x^2 - x^2 + 5x - 6x + 30 = x^3 - 6x^2 - x + 30
\][/tex]
So, the polynomial function of degree 3 with the given zeros [tex]\(-2, 3, 5\)[/tex] is:
[tex]\[
f(x) = x^3 - 6x^2 - x + 30
\][/tex]
