Question

Please help me solve this...

Answer 1

Answer:

$\displaystyle f(x)=\frac{5}{24}(x+2)^2(x-6)(x-2)$

Step-by-step explanation:

The standard, factored polynomial function is given by:

$f(x)=a(x-p)^n(x-q)^m...$

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:

$(x-(-2))=(x+2)$

It is has a multiplicity of 2, it is squared. So:

$(x+2)^2$

We have another root with multiplicity of 1 at x=6.

So, our factor is:

$(x-6)$

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:

$(x-2)$

Therefore, our entire function is:

$f(x)=a(x+2)^2(x-6)(x-2)$

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:

$10=a(0+2)^2(0-6)(0-2)$

Evaluate:

$10=a(4)(-6)(-2)$

Multiply:

$10=48a$

Therefore:

$\displaystyle a=\frac{10}{48}=\frac{5}{24}$

Therefore, our final function is:

$\displaystyle f(x)=\frac{5}{24}(x+2)^2(x-6)(x-2)$