Answer:
[tex]\displaystyle f(x)=\frac{5}{24}(x+2)^2(x-6)(x-2)[/tex]
Step-by-step explanation:
The standard, factored polynomial function is given by:
[tex]f(x)=a(x-p)^n(x-q)^m...[/tex]
Where a is the leading coefficient,
p and q are factors,
And n and m are the powers or multiplicity they are being raised to.
We know that our polynomial function is of degree 4.
We have a root of multiplicity of 2 at x=-2.
So, our factor is:
[tex](x-(-2))=(x+2)\\[/tex]
It is has a multiplicity of 2, it is squared. So:
[tex](x+2)^2[/tex]
We have another root with multiplicity of 1 at x=6.
So, our factor is:
[tex](x-6)[/tex]
And since it is to the first power, we can write it as is.
Finally, we have another root of multiplicity of 1 at x=2.
So, our factor is:
[tex](x-2)[/tex]
Therefore, our entire function is:
[tex]f(x)=a(x+2)^2(x-6)(x-2)[/tex]
We still have to determine our leading coefficient, a.
We can use that y-intercept. The y-intercept is at (0, 10). So, when x=0, y=10. By substitution:
[tex]10=a(0+2)^2(0-6)(0-2)[/tex]
Evaluate:
[tex]10=a(4)(-6)(-2)[/tex]
Multiply:
[tex]10=48a[/tex]
Therefore:
[tex]\displaystyle a=\frac{10}{48}=\frac{5}{24}[/tex]
Therefore, our final function is:
[tex]\displaystyle f(x)=\frac{5}{24}(x+2)^2(x-6)(x-2)[/tex]
