To find the polynomial equation given its zeros, we will work through the problem step-by-step. The provided zeros are:
- 1 with a multiplicity of 2
- [tex]\(2 + \sqrt{2}\)[/tex]
- [tex]\(2 - \sqrt{2}\)[/tex]
## Step 1: Express zeros as factors of the polynomial
Given the zeros, the polynomial can be represented as:
[tex]\[
f(x) = (x - 1)^2 (x - (2 + \sqrt{2}))(x - (2 - \sqrt{2}))
\][/tex]
## Step 2: Simplify the factors
First, we simplify [tex]\((x - 1)^2\)[/tex]:
[tex]\[
(x - 1)^2 = x^2 - 2x + 1
\][/tex]
Next, we simplify [tex]\((x - (2 + \sqrt{2}))(x - (2 - \sqrt{2}))\)[/tex]:
[tex]\[
(x - (2 + \sqrt{2}))(x - (2 - \sqrt{2})) = (x - 2 - \sqrt{2})(x - 2 + \sqrt{2})
\][/tex]
This is a difference of squares:
[tex]\[
= \left((x - 2) - \sqrt{2}\right)\left((x - 2) + \sqrt{2}\right) = (x - 2)^2 - (\sqrt{2})^2 = (x - 2)^2 - 2
\][/tex]
Simplify further:
[tex]\[
(x - 2)^2 - 2 = x^2 - 4x + 4 - 2 = x^2 - 4x + 2
\][/tex]
## Step 3: Multiply the simplified factors to find the polynomial
Now, we multiply the results from Step 2:
[tex]\[
(x^2 - 2x + 1)(x^2 - 4x + 2)
\][/tex]
Use distributive property (FOIL method):
[tex]\[
(x^2 - 2x + 1)(x^2 - 4x + 2) = x^2(x^2 - 4x + 2) - 2x(x^2 - 4x + 2) + 1(x^2 - 4x + 2)
\][/tex]
Distribute [tex]\(x^2\)[/tex]:
[tex]\[
x^2(x^2 - 4x + 2) = x^4 - 4x^3 + 2x^2
\][/tex]
Distribute [tex]\(-2x\)[/tex]:
[tex]\[
-2x(x^2 - 4x + 2) = -2x^3 + 8x^2 - 4x
\][/tex]
Distribute [tex]\(1\)[/tex]:
[tex]\[
1(x^2 - 4x + 2) = x^2 - 4x + 2
\][/tex]
Add all the expanded terms together:
[tex]\[
x^4 - 4x^3 + 2x^2 - 2x^3 + 8x^2 - 4x + x^2 - 4x + 2
\][/tex]
Combine like terms:
[tex]\[
x^4 - 6x^3 + 11x^2 - 8x + 2
\][/tex]
Therefore, the polynomial equation is:
[tex]\[
f(x) = x^4 - 6x^3 + 11x^2 - 8x + 2
\][/tex]
From the given options, the correct choice is:
D. [tex]\(f(x) = x^4 - 6x^3 + 11x^2 - 8x + 2\)[/tex]
