Question

If (x + 1) is a factor of 2x³ + ax² + 2bx + 1, then find the values of a and b given that 2a – 3b = 4.

Answer 1

Given that, (x + 1) is a factor of f(x) = 2x³ + ax² + 2bx + 1, then f(–1) = 0

if (x + a) is a factor of f(x) = ax² + bx + c, then f(–a) = 0

⇒ 2(–1)³ + a(–1)² + 2b(–1) + 1 = 0

⇒ –2 + a – 2b + 1 = 0

⇒ a – 2b – 1 = 0   ......(i)

Also, 2a – 3b = 4

⇒ 3b = 2a – 4

⇒ b = `((2a - 4)/3)`

Now, put the value of b in equation (i), we get

`a - 2((2a - 4)/3) - 1` = 0

⇒ 3a – 2(2a – 4) – 3 = 0

⇒ 3a – 4a + 8 – 3 = 0

⇒ –a + 5 = 0

⇒ a = 5

Now, put the value of a in equation (i), we get

5 – 2b – 1 = 0

⇒ 2b = 4

⇒ b = 2

Hence, the required values of a and b are 5 and 2, respectively.