To construct a polynomial in standard form given its roots, we need to utilize the fact that if [tex]\( r \)[/tex] is a root of the polynomial, then [tex]\( (x - r) \)[/tex] is a factor of the polynomial. The polynomial is the product of these factors.
Given the roots [tex]\( x = 5 \)[/tex] with multiplicity 1 and [tex]\( x = -4 \)[/tex] with multiplicity 2, we construct the polynomial as follows:
### Step-by-Step Solution
1. Identify the Factors:
- For the root [tex]\( x = 5 \)[/tex]: the factor is [tex]\( (x - 5) \)[/tex]
- For the root [tex]\( x = -4 \)[/tex] with multiplicity 2: the factor is [tex]\( (x + 4) \)[/tex] but raised to the power of 2 to account for the multiplicity. Thus, the factor is [tex]\( (x + 4)^2 \)[/tex].
2. Write the Polynomial as a Product of Factors:
[tex]\[
P(x) = (x - 5)(x + 4)^2
\][/tex]
3. Expand the Polynomial:
First, expand [tex]\( (x + 4)^2 \)[/tex]:
[tex]\[
(x + 4)^2 = (x + 4)(x + 4) = x^2 + 8x + 16
\][/tex]
Now, multiply this result by [tex]\( (x - 5) \)[/tex]:
[tex]\[
P(x) = (x - 5)(x^2 + 8x + 16)
\][/tex]
4. Distribute [tex]\( (x - 5) \)[/tex] across [tex]\( (x^2 + 8x + 16) \)[/tex]:
[tex]\[
P(x) = x(x^2 + 8x + 16) - 5(x^2 + 8x + 16)
\][/tex]
Let's distribute [tex]\( x \)[/tex]:
[tex]\[
x(x^2 + 8x + 16) = x^3 + 8x^2 + 16x
\][/tex]
Now, distribute [tex]\( -5 \)[/tex]:
[tex]\[
-5(x^2 + 8x + 16) = -5x^2 - 40x - 80
\][/tex]
Combine the two results together:
[tex]\[
P(x) = x^3 + 8x^2 + 16x - 5x^2 - 40x - 80
\][/tex]
5. Combine Like Terms:
[tex]\[
P(x) = x^3 + 3x^2 - 24x - 80
\][/tex]
### The Polynomial in Standard Form:
[tex]\[
P(x) = x^3 + 3x^2 - 24x - 80
\][/tex]
Thus, the polynomial that fits the given roots and their multiplicities is:
- [tex]\( x = 5 \)[/tex] (multiplicity 1)
- [tex]\( x = -4 \)[/tex] (multiplicity 2)
is
[tex]\[
\boxed{x^3 + 3x^2 - 24x - 80}
\][/tex]
