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Question
Find the value of the constant a and b, if (x – 2) and (x + 3) are both factors of expression x3 + ax2 + bx - 12.
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Solution
Expression x3 + ax2 + bx - 12
(x - 2) is a factor i.e, at x = 2
the remainder will be xero
⇒ (2)3 + a(2)2 + b(2) - 12 = 0
⇒ 8 + 4a + 2b - 12 = 0
⇒ 4a + 2b = 4
⇒ 2a + b = 2 ...(i)
when x + 3 is a factor i.e., at x = -3 the remainder will be zero.
⇒ (-3)3 + a(-3)2 + b(-3) -12 = 0
⇒ -27 + 9a - 3b - 12 = 0
⇒ 9a - 3b = 39
⇒ 3a - b = 13 ...(ii)
Solving (i) and (ii) simultaneously
2a + b = 2
By adding
3a - b = 13
5a = 15
a = 3
Substituting the value of a in the equation (i)
⇒ 2 x 3 + b = 2
⇒ 6 + b = 2
⇒ b = 2 - 6 = -4
⇒ a = 3, b = -4.
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