rational root theorem

Question

Use the rational root theorem to identify the possible roots. Verify with synthetic division and write as a product of its factors. Use the factors to find the remaining roots. Use additional paper for your work.

m(x)=x^4-2x^3-13x^2+14x+24

Answer 1

If p/q is a rational root, then p is a divisor of the constant term, 24 and q is a divisor of the leading coefficient, 1.

So, the possibilities are 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 8, -8, 12, -12, 24, -24.

By trial and error, we see that -1 is a root, so x - (-1) = x + 1 is a factor

Divide out by x + 1 and get m(x) = (x + 1)(x³ - 3x² - 10x + 24)

Again by trial and error, 2 is a root of x³ - 3x² - 10x + 24. Dividing by x - 2, we then have:

m(x) = (x + 1)(x - 2)(x² - x - 12)

Factoring the quadratic factor gives m(x) = (x + 1)(x - 2)(x - 4)(x + 3)