If (x + 2) and (x + 3) are factors of x3 + ax + b, find the values of

1.	If (x + 2) and (x + 3) are factors of x3 + ax + b, find the values of ‘a’ and ‘b’.
Answer» Given that (x + 2) and (x + 3) are factors of p(x) = x³ + ax + b. ∴ p(- 2) = (- 2)³+ 0(- 2) + b = 0 ⇒ – 8 – 2a + b = 0 => – 2a + b = 8 …….(i) And p(- 3) = (- 3)³ + a(- 3) + b = 0 ⇒ – 27 – 3a + b = 0 => – 3a + b = 27 ……..(ii) Subtracting (i) from (ii), we obtain (-3a 4 – b) – (-2a + b) = 27 – 8 – 3a + b + 2a – b = 19 -a = 19 ⇒ a = 19 From (i), we obtain – 2(19) + b = 8 – 38 + b = 8 ⇒ b = 8 + 38 ⇒ b = 46 Hence, the values of a and b are a = 19 and b = 46.

If (x + 2) and (x + 3) are factors of x3 + ax + b, find the values of ‘a’ and ‘b’.

Answer»

Given that (x + 2) and (x + 3) are factors of p(x) = x³ + ax + b.

∴ p(- 2) = (- 2)³+ 0(- 2) + b = 0

⇒ – 8 – 2a + b = 0 => – 2a + b = 8 …….(i)

And p(- 3) = (- 3)³ + a(- 3) + b = 0

⇒ – 27 – 3a + b = 0 => – 3a + b = 27 ……..(ii)

Subtracting (i) from (ii), we obtain

(-3a 4 – b) – (-2a + b) = 27 – 8

– 3a + b + 2a – b = 19

-a = 19

⇒ a = 19

From (i), we obtain

– 2(19) + b = 8

– 38 + b = 8

⇒ b = 8 + 38

⇒ b = 46

Hence, the values of a and b are a = 19 and b = 46.