Question

determine the roots algebraically by factoring.
\[4x^4-4x^3-51x^2+106x-40\]

Answer 1

Do you know synthetic division?

Answer 2

Thanks to the rational root theorem, we have some candidates to try for roots. Factor -40, and 4 (the constant term and the coefficient of the leading term respectively). Any rational roots will be found on a list made up of all the different factors of the constant term over all the different factors of the leading term. There are quite a few of them here!

You can test a root candidate by plugging it into the polynomial. If the polynomial evaluates to 0, that candidate is a root.

With each root \(r_n\) you find, you can divide the polynomial by \((x-r_n)\) to get a simpler polynomial with the same roots as the roots you have yet to find in the original.

Answer 3

@Mertsj yes i do

Answer 4

The possible shelf numbers are the factors of the constant term, which is 40, divided by the coefficient of the leading term, which is 4.

Answer 5

So here are the possibilities:
\[\frac{\pm 1,\pm 2,pmn4,\pm 5, \pm8, \pm10, \pm20, \pm40}{\pm 1, \pm 2, \pm4}\]

Answer 6

Test those until you find one which gives a remainder of 0.

Answer 7

we could make this a probability question: what is the probability that any root candidate, randomly selected, is actually a root of P(x)? :-)