Question

What first degree polynomial added to 3x³ − 2x² gives a polynomial for which both x − 1 and x + 1 are factors?

Answer 1

Let the first degree polynomial that has to be added to the polynomial 3x³− 2x² be ax + b.

So, the polynomial is 3x³− 2x² + ax + b.

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

⇒ 3 − 2 + a + b = 0

⇒ a + b = −1 …(1)

Similarly, checking for (x + 1).

(x + 1) = {x − (−1)}

We must have:

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

⇒ −3 − 2 − a + b = 0

⇒ b − a = 5 …(2)

Adding (1) and (2):

2b = 4

⇒ b = 2

a = −3

Therefore, the first degree expression that has to be added to 3x³− 2x² is −3x + 2, so that the polynomial obtained has both (x + 1) and (x − 1) as the factors.