Question

Consider the polynomial f(x) = 3x3 – 2x2 – 7x – 2.

(a) By using the Rational Zero Theorem, list all possible rational zeros of the given polynomial.

(b) Find all of the zeros of the given polynomial. Be sure to show work, explaining how you have found them.

Answer 1

I should try x = ±1, ±2, ±2/3, and ±1/3

f(1) = 3-2-7-2 ≠ 0
f(-1) = -3-2+7-2 = 0 , so x+1 is a factor
f(-2) = -24 -8 + 14 - 2 ≠0
f(2) = 24 - 8 - 14 - 2 = 0 , so x-2 is a factor
f(2/3) = 8/9 - 8/9 - 14/3 - 2 ≠ 0
f(-2/3) = -8/9 - 8/9 + 14/3 - 2 ≠0
f(1/3) = 1/9 - 2/9 - 7/9 - 2 ≠ 0
f(-1/3) = -1/9 - 2/9 + 7/3 - 2 = 0 , so (3x+1) is a factor


3x^3 – 2x^2 – 7x – 2 = (3x+1)(x-2)(x+1)

x = -1, -1/3, and 2