To find a polynomial [tex]\( P(x) \)[/tex] of degree 5 that has specific roots and multiplicities, we follow these steps:
1. Identify the roots and their multiplicities:
- A root at [tex]\( x = 2 \)[/tex] with multiplicity 2.
- A root at [tex]\( x = 0 \)[/tex] with multiplicity 2.
- A root at [tex]\( x = -2 \)[/tex] with multiplicity 1.
2. Construct the polynomial using the identified roots and their multiplicities:
- The factor corresponding to the root [tex]\( x = 2 \)[/tex] with multiplicity 2 is [tex]\( (x - 2)^2 \)[/tex].
- The factor corresponding to the root [tex]\( x = 0 \)[/tex] with multiplicity 2 is [tex]\( x^2 \)[/tex].
- The factor corresponding to the root [tex]\( x = -2 \)[/tex] with multiplicity 1 is [tex]\( (x + 2) \)[/tex].
3. Combine all the factors to form the polynomial:
[tex]\[
P(x) = (x - 2)^2 \cdot x^2 \cdot (x + 2)
\][/tex]
4. Expand the polynomial to express it in standard form:
- First, expand [tex]\( (x - 2)^2 \)[/tex]:
[tex]\[
(x - 2)^2 = x^2 - 4x + 4
\][/tex]
- Next, include the [tex]\( x^2 \)[/tex] factor:
[tex]\[
P(x) = x^2 \cdot (x^2 - 4x + 4)
\][/tex]
This gives:
[tex]\[
P(x) = x^4 - 4x^3 + 4x^2
\][/tex]
- Finally, multiply by the [tex]\( (x + 2) \)[/tex] factor:
[tex]\[
P(x) = (x^4 - 4x^3 + 4x^2)(x + 2)
\][/tex]
- Expand this product:
[tex]\[
P(x) = x^5 + 2x^4 - 4x^4 - 8x^3 + 4x^3 + 8x^2
\][/tex]
Combine like terms:
[tex]\[
P(x) = x^5 - 2x^4 - 4x^3 + 8x^2
\][/tex]
Thus, the polynomial [tex]\( P(x) \)[/tex] that satisfies the given conditions is:
[tex]\[
P(x) = x^5 - 2x^4 - 4x^3 + 8x^2
\][/tex]
