Question

What number added to 3x³ − 2x² + 5x given a polynomial for which x − 1 is a factor?

Answer 1

Let the number that has to be added to the polynomial 3x³− 2x² + 5x be c.

So, the polynomial is 3x³− 2x² + 5x + c.

Divisor = (x − 1)

Factor theorem says that for the polynomial p(x) and for the number a, if we have p(a) = 0, then (x − a) is a factor of p(x).

Thus, we must have:

3(1)³ − 2(1)² + 5(1) + c = 0 for (x − 1) to be a factor of 3x³− 2x² + 5x + c.

⇒ 3 − 2 + 5 + c = 0

⇒ c = −6

Thus, −6 must be added to 3x³− 2x² + 5x to obtain a polynomial which has (x − 1) as a factor.