Let's construct the polynomial function \( f(x) \) given the specified roots and leading coefficient. We'll go through the process step-by-step:
1. Identify the roots and leading coefficient:
- Given roots: \( 5 \), \( \sqrt{5} \), and \( -\sqrt{5} \)
- Leading coefficient: \( 4 \)
2. Write the polynomial in factored form using the roots:
A root \( r \) of a polynomial can be represented by the factor \( (x - r) \). Therefore, the polynomial with the given roots can be expressed as:
[tex]\[
f(x) = 4(x - 5)(x - \sqrt{5})(x + \sqrt{5})
\][/tex]
3. Simplify the factored form:
Let's first deal with the two roots involving square roots. Notice:
[tex]\[
(x - \sqrt{5})(x + \sqrt{5}) = x^2 - (\sqrt{5})^2 = x^2 - 5
\][/tex]
Substituting back into the equation, we have:
[tex]\[
f(x) = 4(x - 5)(x^2 - 5)
\][/tex]
4. Expand the expression:
We'll now expand \( (x - 5)(x^2 - 5) \):
[tex]\[
(x - 5)(x^2 - 5) = x(x^2 - 5) - 5(x^2 - 5)
\][/tex]
[tex]\[
= x^3 - 5x - 5x^2 + 25
\][/tex]
Reordering terms for clarity:
[tex]\[
= x^3 - 5x^2 - 5x + 25
\][/tex]
5. Apply the leading coefficient:
Finally, multiply the expanded polynomial by the leading coefficient \( 4 \):
[tex]\[
f(x) = 4(x^3 - 5x^2 - 5x + 25)
\][/tex]
[tex]\[
= 4x^3 - 20x^2 - 20x + 100
\][/tex]
So, the polynomial function \( f(x) \) in standard form is:
[tex]\[
f(x) = 4x^3 - 20x^2 - 20x + 100
\][/tex]
This is the polynomial function having a leading coefficient of 4 and the roots [tex]\( 5 \)[/tex], [tex]\( \sqrt{5} \)[/tex], and [tex]\( -\sqrt{5} \)[/tex].
