If x + 1 is a factor of 2x³ + ax² + 2bx + 1, and 2a – 3b = 4, then the value of a + 2b is:
Polynomial: 2x³ + ax² + 2bx + 1
x + 1 is a factor → f(–1) = 0
Also given: 2a – 3b = 4
Formula used:
If x + 1 is a factor, then f(–1) = 0
Solution:
f(–1) = 2(–1)³ + a(–1)² + 2b(–1) + 1 = 0
=> –2 + a – 2b + 1 = 0
=> a – 2b = 1 ...(1)
Also given:
2a – 3b = 4 ...(2)
From (1):
a = 2b + 1
Substitute into (2):
2(2b + 1) – 3b = 4
=> 4b + 2 – 3b = 4
=> b + 2 = 4
=> b = 2
Now,
a = 2(2) + 1 = 5
a + 2b = 5 + 4 = 9
Correct answer is (c)
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