# How to find an unknown in a cubic equation iF you are given a factor?

1. Mar 31, 2012

### Beurre

An example is x^3 + x^2 + ax -72
Factor is x+3
F(-3) doesnt equal zero and im out of other ideas. Help? :/

2. Mar 31, 2012

### jing2178

If x+3 is a factor then F(-3) = 0. Why do you think F(-3) is not 0? What would make it 0?

3. Mar 31, 2012

### robert barron

thats the problem, think without any exemple.

4. Apr 1, 2012

### SammyS

Staff Emeritus
Hello Beurre. Welcome to PF !

What is F(-3) ?

5. Apr 1, 2012

### Bearded Man

Have you learned the relationship between roots and coefficients?
$$S_n = x_1x_2...x_n = (-1)^n \dfrac{a_0}{a_n}$$
That is, the sum of the roots taken n at a time (in all possible combinations) equals the constant term divided by the nth coefficient multiplied by negative one raised to the nth power. I encourage you to research why this is true, so you don't blindly use the theorem. Regardless, let
$$P(x) \textrm{ have roots } x_1, x_2, \textrm{ and } x_3$$
Then $$S_1 = x_1 + x_2 + x_3 ; S_2 = x_1x_2 + x_1x_3 + x_2x_3 ; S_3 = x_1x_2x_3$$

If not, and you are given at least one root and there is one coefficient missing, you can do synthetic division with x = -3 and deduce what that value must be.
I can't really type synthetic division out here, but try doing it, because you know that the some value times a must equal 72.

Last edited: Apr 1, 2012
6. Apr 1, 2012

### HallsofIvy

Staff Emeritus
Beurre don't seem to have got back to us.

Beurre, have you been able to do this? If not, since you assert that "F(-3) doesnt equal zero", what do you think it does equal?

7. Apr 1, 2012

### rcgldr

You could do long hand polynomial division of the cubic equation by the known factor in order to end up with a quadratic equation and a remainder that will be some linear function of a, which you can then solve for a, or as already suggested subsitute x = -3 into the equation and solve for a.