Question

Find the rational roots of x^4+8x^3+7x^2-40x-60=0

Answer 1

@hartnn

Answer 2

You can use the rational roots theorem to guess and check the possible roots. You will find that (x + 2) and (x + 6) are factors of this polynomial. The other factor is (x^2 - 5). Can you obtain the rational roots from here?

Answer 3

yes but how did you get those numbers

Answer 4

Using the Rational Roots Theorem ( http://www.mathwords.com/r/rational_root_theorem.htm) you guess and check the possible values. The possible rational roots are +- p/q where p are the factors of 60 and q are the factors of 1. I.e., the possible rational roots are +- 1, +-2, +-4, +-5, +-6., etc... By guess and check we can try x = 1, x = -1, x = -2, x = 2 and x = -2. We find that x = -2 results in a 0. That means (x + 2) is a factor. Divide the original polynomial by (x + 2) to obtain a simpler polynomial of degree 3 and repeat the same steps to obtain the next factor.

Answer 5

ok

Answer 6

I repeated x = -2 twice by accident in the explanation above.

Anyway, to show x = -2 results in a 0:

(-2)^4+8(-2)^3+7(-2)^2-40(-2)-60 =
16 - 64 + 28 + 80 - 60 = 0

So if x = -2, that means (x + 2) must be a factor of the original polynomial. Division yields (see attached file).

Our new polynomial is x^3 + 6x^2 - 5x - 30. We can use the same Rational Roots Theorem to determine the remaining possible rational root(s).