factoring polynomial

Question

using the factors (3x + 2) & (x - 1), find the remaining factor(s) of

f(x) = -6x⁴ + 23x³ - 12x² - 11x + 6 and write polynomial in fully factored form.

Answer 1

You could also do this with one division by dividing (long division) the given 4th degree polynomial by the product of the 2 given factors, which would be 3x² - x - 2, and give -2x² + 7x - 3 and the same factors as above. This may be somewhat easier than performing 2 divisions.

Answer 2

One way to determine the other factors is to divide these two given factors into -6x⁴ + 23x³ - 12x² - 11x + 6 and see what is left. To do this, we can first use synthetic division with x - 1 as follows:

Hopefully you know how to do this. If not, send me a message and I'll walk you through it.

That leaves -6x³ + 17x² + 5x - 6. Now, do long division for 3x + 2 as follows:

This leaves -2x² + 7x - 3 which factors into (-2x +1)(x - 3)

So f(x) = (3x + 2)(x - 1)(-2x + 1)(x - 3)