Mathematics
If a polynomial x3 + 2x2 – ax + b leaves a remainder -6 when divided by x + 1 and the same polynomial has x - 2 as a factor, then find the values of a and b.
Factorisation
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Answer
Given,
x3 + 2x2 – ax + b leaves a remainder -6 when divided by x + 1.
⇒ x + 1 = 0
⇒ x = -1.
Let f(x) = x3 + 2x2 – ax + b, then f(-1) = -6.
⇒ f(-1) = -6
⇒ (-1)3 + 2(-1)2 - a(-1) + b = -6
⇒ -1 + 2(1) + a + b = -6
⇒ -1 + 2 + a + b = -6
⇒ a + b + 1 = -6
⇒ a + b = -6 - 1
⇒ a + b = -7 ……..(1)
Given ,
x - 2 is a factor of f(x).
⇒ x - 2 = 0
⇒ x = 2.
f(2) = 0
⇒ 23 + 2(2)2 – 2a + b = 0
⇒ 8 + 2(4) - 2a + b = 0
⇒ 8 + 8 - 2a + b = 0
⇒ 16 - 2a + b = 0
⇒ 2a - b = 16 ……(2)
Adding equation (1) and (2), we get :
⇒ a + b + 2a - b = -7 + 16
⇒ 3a = 9
⇒ a = = 3.
Substituting value of a = 3 in equation (1), we get :
⇒ 3 + b = -7
⇒ b = -7 - 3 = -10.
Hence, a = 3 and b = -10.
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