Final answer:
The third factor of the polynomial f(x) = 3x^2 + 13x - 8, which is obtained after correcting the given polynomial, is (x - 2), found by dividing the polynomial by the product of the given factors (3x + 1) and (x + 4).
Step-by-step explanation:
To find the third factor of the polynomial f(x) = 3x2 + 7x2 - 22x - 8, we first need to correct the polynomial, as there seems to be a typo since 3x2 and 7x2 are both terms with x squared. I suspect the correct polynomial should be f(x) = 3x2 + 13x - 8, which is formed by the product of its factors (3x + 1) and (x + 4). Multiplying these two factors, we get:
(3x + 1)(x + 4) = 3x2 + 12x + x + 4 = 3x2 + 13x + 4
To find the third factor, we need to divide the corrected polynomial by this product:
(3x2 + 13x - 8) ÷ (3x2 + 13x + 4) = (x - 2)
Therefore, the third factor is option a) (x - 2).