How to get the X-intercept

[Dowd0023]

How would I go about getting the X-intercept to get the critical values to plot.

F(x)=1/2(x^4-11x^3+28x^2)

 

[MarkFL]

Hello, and welcome to FMH!

Whenever you're trying to get the roots of a polynomial function, you want to try to factor. Do you see a factor common to all terms within the parentheses?

 

[pka]

How would I go about getting the X-intercept to get the critical values to plot.
F(x)=1/2(x^4-11x^3+28x^2)

You should know that \(\displaystyle x^4-11x^3+28x^2=x^2(x^2-11x+28)=x^2(x-7)(x-4)=0\)

 

[Dowd0023]

Thanks that is what I thought it was but I was not sure what you do with that 1/2 outside?

 

[Dowd0023]

X^2 would be a factor

 

[Dr.Peterson]

Thanks that is what I thought it was but I was not sure what you do with that 1/2 outside?

That has no effect on x-intercepts; all it does is stretch the graph of the function vertically.

X^2 would be a factor

Yes; so the corresponding intercept has multiplicity 2. Do you know what that tells you about the graph?

Each (linear) factor provides one (real) zero of the function, and therefore one x-intercept. Higher multiplicity just changes its behavior near there.