Enter the polynomial function [tex]$f(x)[tex]$[/tex] in standard form given that [tex]$[/tex]f[tex]$[/tex] has a leading coefficient of 4 and roots [tex]$[/tex]5, \sqrt{5}[tex]$[/tex], and [tex]$[/tex]-\sqrt{5}$[/tex].



Answer :

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].