Finding rational zeros: x^4 - (3/2)x^3 - 7x^2 + 9x +6

[zooanimal98]

I have this math problem:

Find the rational zeros:

x^4 - (3/2)x^3 - 7x^2 + 9x +6

I am pretty sure the answer is 2 since I used my calculator, but I would like know how to solve it by factoring.

 

[tkhunny]

Re: Finding rational zeros

"Pretty sure"? What does that mean?

Multiply it by 2.

The 2 on the x^4 term is important.
The constant 12 is important.

Using just the 12, we have 1,2,3,4,6,12 as possible Rational Roots (Positive or negative)
Combining with the only factor besides 1 from the 2, we add only 1/2, 3/2 (Positive or Negative)

This is an excellent place to start. 1/2, 1, 3/2, 2, 3, 4, 6, and 12 (positive or negative) are the ONLY possible rational roots.

There are various ways to narrow the list down some more, but in this case we believe already that x = 2 works. How is your synthetic division? That would be the easiest way to check and to reduce your polynomial by one degree. Let's see what you get.

 

[Deleted member 4993]

Re: Finding rational zeros

You have one other rational root - and two other real root (probably irrational) around ±2.5