How do you use the rational root theorem to find the roots of 2x^4 + 3x^3 - 7x^2 + 3x - 9 = 0 ?

1 Answer
May 23, 2015

Any rational root p/q - if written in lowest terms, so that p and q have no common factor other than 1, satisfies the property that p is a divisor of the constant term -9 and q is a divisor of the coefficient 2 of the highest order term (2x^4).

So the only possible rational roots are:

+-1/2, +-1, +-3/2, +-3, +-9/2 or +-9.

Rustling up a quick spreadsheet to help, I found that -3 and 3/2 are roots, hence (x+3) and (2x-3) are factors of 2x^4+3x^3-7x^2+3x-9.

(x+3)(2x-3) = (2x^2+3x-9)

Use synthetic division to find:

2x^4+3x^3-7x^2+3x-9 = (2x^2+3x-9)(x^2+1)

So the other 2 roots of 2x^4+3x^3-7x^2+3x-9 = 0 are +-i