Function Question

[dagr8est]

If f(x)=x^4−3x^3−9x^2+4, for how many real numbers k does f(k)=2?

a)none
b)one
c)two
d)three
e)four

I'm sure I've learned how to solve these types of questions before, but I just can't remember the process. I think it hard something to do with synthetic division. Can someone explain how to solve this question?

 

[dtoepel]

Do you have a graphing calculator? If you graph this function along with the graph of the line y = 2 you should see the correct answer.

Dave

 

[Matt]

If f(x)=x^4−3x^3−9x^2+4, for how many real numbers k does f(k)=2?

a)none
b)one
c)two
d)three
e)four

Find the roots of g(x) = x⁴ - 3x³ - 9x² + 2.

 

[dagr8est]

That's what I tried to do before posting this question. I formed the equation

0=x^4−3x^3−9x^2+2

and tried to solve for the roots using synthetic division. The roots of that function aren't integers though so synthetic division didn't work. That's why I thought I wasn't approaching this problem properly.

I did it on a graphing calculator, and there are 4 non-integer roots of that function and yes the correct answer is e)4. Is there anyway to find the roots of a quartic equation which has non-integer roots without using a graphing calculator?

 

[stapel]

Is there anyway to find the roots of a quartic equation which has non-integer roots without using a graphing calculator?

Yeah, but... you won't like it.

Are you pretty sure that you're not allowed to use the graph? Or could you, working from the graph, pick x-values such that f(x₁) > 0, f(x₂) < 0, f(x₃) > 0, f(x₄) < 0, and f(x₅) > 0, and then use the fact that polynomials are continuous to show that, while you may not know what the exact zeroes are, they have to lie between the five listed x-values...?

Eliz.

 

[dagr8est]

You're right, that is messy. I'll just stick with the graphing calculator solution then. Thanks for the help.