Find the values of a and b so that (2x^3 + a x^2 + x + b) has (x + 2) and (2x – 1) as factors.
Let P(x) = 2x^3 + a x^2 + x + b.
Since (x + 2) is a factor then x = -2 is a root:
P(-2) = 2(-8) + a(4) + (-2) + b = -16 + 4a - 2 + b = 0
=> 4a + b - 18 = 0
=> b = 18 - 4a ...(1)
Since (2x – 1) is a factor then x = 1/2 is a root:
P(1/2) = 2(1/8) + a(1/4) + 1/2 + b = 1/4 + a/4 + 1/2 + b = 0
=> a/4 + b + 3/4 = 0
=> b = -3/4 - a/4 ...(2)
Equate (1) and (2):
18 - 4a = -3/4 - a/4
Multiply by 4: 72 - 16a = -3 - a
=> 72 + 3 = 16a - a
=> 75 = 15a
=> a = 5
From (1): b = 18 - 4a = 18 - 20 = -2
Answer:
a = 5, b = -2
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