Sure! Let's find the polynomial [tex]\( P(x) \)[/tex] that satisfies the given conditions.
### Step-by-Step Solution:
1. Identify the Roots and Their Multiplicities:
- The polynomial has a root of multiplicity 2 at [tex]\( x = 1 \)[/tex].
- The polynomial has a root of multiplicity 2 at [tex]\( x = 0 \)[/tex].
- The polynomial has a root of multiplicity 1 at [tex]\( x = -3 \)[/tex].
2. Express the Polynomial in Factored Form:
Based on the given roots and their multiplicities, we can write the polynomial in its factored form as:
[tex]\[
P(x) = (x - 1)^2 \cdot x^2 \cdot (x + 3)
\][/tex]
3. Expand the Factors to Form the Polynomial:
We need to multiply the factors together and expand them to get the polynomial in standard form.
[tex]\[
P(x) = (x - 1)^2 \cdot x^2 \cdot (x + 3)
\][/tex]
First, expand [tex]\((x - 1)^2\)[/tex]:
[tex]\[
(x - 1)^2 = x^2 - 2x + 1
\][/tex]
Now, include [tex]\( x^2 \)[/tex]:
[tex]\[
(x^2 - 2x + 1) \cdot x^2 = x^4 - 2x^3 + x^2
\][/tex]
Finally, multiply by [tex]\((x + 3)\)[/tex]:
[tex]\[
(x^4 - 2x^3 + x^2) \cdot (x + 3)
\][/tex]
Distribute each term of the first polynomial by each term of the second polynomial:
[tex]\[
x^4 \cdot (x + 3) = x^5 + 3x^4
\][/tex]
[tex]\[
-2x^3 \cdot (x + 3) = -2x^4 - 6x^3
\][/tex]
[tex]\[
x^2 \cdot (x + 3) = x^3 + 3x^2
\][/tex]
Now, combine all these terms together:
[tex]\[
x^5 + 3x^4 - 2x^4 - 6x^3 + x^3 + 3x^2
\][/tex]
4. Combine Like Terms:
[tex]\[
x^5 + (3x^4 - 2x^4) + (-6x^3 + x^3) + 3x^2
\][/tex]
[tex]\[
x^5 + x^4 - 5x^3 + 3x^2
\][/tex]
Thus, the polynomial [tex]\( P(x) \)[/tex] is:
[tex]\[
P(x) = x^5 + x^4 - 5x^3 + 3x^2
\][/tex]
This polynomial has a degree of 5, a leading coefficient of 1, and the given roots with their respective multiplicities.
