Question

One root of f(x) = 2x3 + 9x2 + 7x – 6 is –3. Explain how to find the factors of the polynomial.

Answer 1

Use the Factor Theorem: if x = r is a root then (x − r) is a factor. First check f(−3):

f(−3) = 2(−3)^3 + 9(−3)^2 + 7(−3) − 6 = 2(−27) + 9(9) − 21 − 6 = −54 + 81 − 21 − 6 = 0,

so (x + 3) is a factor. Divide the cubic by (x + 3) (synthetic division is quick):

Coefficients 2, 9, 7, −6 with root −3:
bring down 2
−3·2 = −6, 9 + (−6) = 3
−3·3 = −9, 7 + (−9) = −2
−3·(−2) = 6, −6 + 6 = 0

Quotient is 2x^2 + 3x − 2. Factor that quadratic:
2x^2 + 3x − 2 = (2x − 1)(x + 2).

So the full factorization is (x + 3)(2x − 1)(x + 2), and the roots are x = −3, 1/2, −2.