To find the polynomial function with the specified properties, we'll follow these steps:
1. Identify the zeros and their multiplicities:
- A zero at [tex]\( x = -4 \)[/tex] with multiplicity 3 means that [tex]\( (x + 4) \)[/tex] appears three times as a factor.
- A zero at [tex]\( x = 0 \)[/tex] with multiplicity 1 means that [tex]\( x \)[/tex] appears once as a factor.
2. Form the polynomial using these factors:
- For [tex]\( x = -4 \)[/tex] with multiplicity 3, the factor is [tex]\( (x + 4)^3 \)[/tex].
- For [tex]\( x = 0 \)[/tex] with multiplicity 1, the factor is [tex]\( x \)[/tex].
3. Combine these factors to form the polynomial:
[tex]\[
f(x) = x \cdot (x + 4)^3
\][/tex]
4. Expand the polynomial to express it in standard form:
First, expand [tex]\( (x + 4)^3 \)[/tex]. Using the binomial theorem:
[tex]\[
(x + 4)^3 = x^3 + 3 \cdot 4x^2 + 3 \cdot 4^2x + 4^3
\][/tex]
[tex]\[
(x + 4)^3 = x^3 + 12x^2 + 48x + 64
\][/tex]
Next, multiply this result by [tex]\( x \)[/tex]:
[tex]\[
f(x) = x \cdot (x^3 + 12x^2 + 48x + 64)
\][/tex]
[tex]\[
f(x) = x^4 + 12x^3 + 48x^2 + 64x
\][/tex]
Therefore, the polynomial function in expanded form is:
[tex]\[
f(x) = x^4 + 12x^3 + 48x^2 + 64x
\][/tex]
This is the polynomial of degree 4 with the given zeros and their specified multiplicities.
