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The polynomial of degree 5, [tex]\( P(x) \)[/tex], has a leading coefficient of 1, roots of multiplicity 2 at [tex]\( x=2 \)[/tex] and [tex]\( x=0 \)[/tex], and a root of multiplicity 1 at [tex]\( x=-2 \)[/tex].

Find a possible formula for [tex]\( P(x) \)[/tex].

[tex]\[ P(x) = \][/tex]
[tex]\[ \square \][/tex]


Sagot :

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]
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