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

Algebra (10)

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

Answer

$$ (x + 2) \text{and} (x + 3) \text{factors} \to  x^3 + ax + b $$

$$ f(-2) = (-2)^3 + a (-2) + b $$

$$ f(-3) = (-3)3 + a(-3) + b $$

$$ 0 = -8 -2a + b $$

$$ -2a + b = 8 $$

$$ 0 = -27 -3a + b $$

$$ –3a + b = 27 $$

$$ –2a + b = 8 $$

$$ –3a + b = 27 $$

$$ a = -19 $$

$$ 38 + b = 8 $$

$$ b = -30 $$