Question

If (x + 1) and (x + 3) are the factors of x³ + ax + b, find the values of a and b.

Answer 1

Let f(x) = x³ + ax + b

Let x + 1 = 0

x = −1

∵ x + 1 is a factor of f(x).

∴ f(−1) = 0

⇒ (−1)³ + a(−1) + b = 0

⇒ −1 − a + b = 0

⇒ b − a = 1   ...(i)

∵ x + 3 is a factor of f(x).

∴ f(−3) = 0

⇒ (−3)³ + a(−3) + b = 0

⇒ −27 − 3a + b = 0

⇒ b − 3a = 27   ...(ii)

Subtract (i) from (ii),

(b − 3a) − (b − a) = 27 − 1

−3a + a = 26

−2a = 26

a = −13

Now substitute a = −13 in equation (i)

b − (−13) = 1

b + 13 = 1

b = 1 − 13

b = −12

Hence, a = −13 and b = –12.