# How to calculate zeros withour graphing technology?

• May 15th 2011, 06:47 AM
Barthayn
How to calculate zeros withour graphing technology?
The function f(x) = 3x^4 + 2x^3 - 15x^2 + 12x -2 is not able to factor without graphing technology because you can only find that the zero is x = 1 without graphing technology.

Is there a method that I am missing for using test points on the cubic function. I used all plus/minus 0.5,1,2.
• May 15th 2011, 07:05 AM
Quacky
Quote:

Originally Posted by Barthayn
The function f(x) = 3x^4 + 2x^3 - 15x^2 + 12x -2 is not able to factor without graphing technology because you can only find that the zero is x = 1 without graphing technology.

Is there a method that I am missing for using test points on the cubic function. I used all plus/minus 0.5,1,2.

I suppose you could use Descartes' rule of signs to find an approximation for the number of roots.

But you are really asking about the rational roots test, I think, which is a small expansion of the factor theorem.

You have to consider that some factors might be of the form $\displaystyle (3x\pm k)$, where 'k' refers to any of the factors of $\displaystyle 2$, by testing $\displaystyle f(\frac{-k}{3})$, as you would with the factor theorem. You should find that there are indeed more factors to this polynomial.(Wink)
• May 15th 2011, 07:41 AM
Barthayn
Quote:

Originally Posted by Quacky
I suppose you could use Descartes' rule of signs to find an approximation for the number of roots.

But you are really asking about the rational roots test, I think, which is a small expansion of the factor theorem.

You have to consider that some factors might be of the form $\displaystyle (3x\pm k)$, where 'k' refers to any of the factors of $\displaystyle 2$, by testing $\displaystyle f(\frac{-k}{3})$, as you would with the factor theorem. You should find that there are indeed more factors to this polynomial.(Wink)

I know that there are more factors, however, the zeros are x = -2.89681, x = 0.23013, and x = 1.

The -2.89 and 0.23 zeros cannot come from any test points that one can find. Therefore, one must use graphing technology, correct?

EDIT: Nevermind, I forgot that there was a (x-1)^2 on this function. I feel like a mathematical fool. :(