To write a polynomial [tex]\( f(x) \)[/tex] that meets the specified conditions, follow these steps:
1. Identify the roots and their multiplicities:
- Root [tex]\( 3 \)[/tex] with multiplicity 1
- Root [tex]\( \frac{7}{2} \)[/tex] with multiplicity 1
- Root [tex]\( 0 \)[/tex] with multiplicity 4
2. Express the polynomial in factored form:
Using the roots and their multiplicities, we can write the polynomial as:
[tex]\[
f(x) = (x - 3)^1 (x - \frac{7}{2})^1 (x - 0)^4
\][/tex]
This can be simplified to:
[tex]\[
f(x) = (x - 3) (x - \frac{7}{2}) x^4
\][/tex]
3. Expand the polynomial:
We need to multiply these factors together and expand them to get the polynomial in standard form.
First, simplify the term with fractional coefficient:
[tex]\[
(x - \frac{7}{2}) = (x - \frac{7}{2}) = \left(x - \frac{7}{2}\right)
\][/tex]
Rewrite it as:
[tex]\[
f(x) = x^4 (x-3)(x-\frac{7}{2})
\][/tex]
4. Combine [tex]\( (x - 3) \)[/tex] and [tex]\( (x - \frac{7}{2}) \)[/tex] first:
[tex]\[
(x - 3)(x - \frac{7}{2}) = x^2 - 3x - \frac{7}{2}x + \frac{21}{2}
\][/tex]
Combine like terms:
[tex]\[
x^2 - \frac{13}{2}x + \frac{21}{2}
\][/tex]
5. Multiply by [tex]\( x^4 \)[/tex]:
[tex]\[
x^4 \left( x^2 - \frac{13}{2}x + \frac{21}{2} \right)
\][/tex]
Distribute [tex]\( x^4 \)[/tex]:
[tex]\[
= x^6 - \frac{13}{2}x^5 + \frac{21}{2}x^4
\][/tex]
So, the polynomial [tex]\( f(x) \)[/tex] in standard form that meets the given conditions is:
[tex]\[
f(x) = x^6 - \frac{13}{2}x^5 + \frac{21}{2}x^4
\][/tex]
