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Mathematics

Find the value of the constants a and b, if (x - 2) and (x + 3) are both factors of the expression x3 + ax2 + bx - 12.

Factorisation

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Answer

f(x) = x3 + ax2 + bx - 12

If (x - 2) and (x + 3) or (x - (-3)) are factors of f(x) then, f(2) and f(-3) = 0.

∴f(2)=23+a(2)2+2b−12=0⇒8+4a+2b−12=0⇒4a+2b−4=0⇒4a+2b=4\therefore f(2) = 2^3 + a(2)^2 + 2b - 12 = 0 \\[0.5em] \Rightarrow 8 + 4a + 2b - 12 = 0 \\[0.5em] \Rightarrow 4a + 2b - 4 = 0 \\[0.5em] \Rightarrow 4a + 2b = 4

On dividing equation by 2,

⇒2a+b=2b=2−2a  (Equation 1)\Rightarrow 2a + b = 2 \\[0.5em] b = 2 - 2a \text{ \space (Equation 1)}

∴f(−3)=(−3)3+a(−3)2+(−3)b−12=0⇒−27+9a−3b−12=0⇒9a−3b−39=0\therefore f(-3) = (-3)^3 + a(-3)^2 + (-3)b - 12 = 0 \\[0.5em] \Rightarrow -27 + 9a - 3b - 12 = 0 \\[0.5em] \Rightarrow 9a - 3b - 39 = 0

Putting value of b = 2 - 2a from equation 1,

⇒9a−3(2−2a)−39=0⇒9a−6+6a−39=0⇒15a−45=0⇒15a=45⇒a=3∴b=2−2a=2−6=−4\Rightarrow 9a - 3(2 - 2a) - 39 = 0 \\[0.5em] \Rightarrow 9a - 6 + 6a - 39 = 0 \\[0.5em] \Rightarrow 15a - 45 = 0 \\[0.5em] \Rightarrow 15a = 45 \\[0.5em] \Rightarrow a = 3 \\[0.5em] \therefore b = 2 - 2a = 2 - 6 = -4

Hence, the value of a is 3 and that of b is -4.

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