Question

Find the value of a and b so that (2x³+ax²+x+b) has (x+2) and (2x-1) as a factors.

Answer 1

Answer:

a = 5, b = -2

Step-by-step explanation:

(x+2) and (2 x-1) is factor of 2 x³+ a x²+ x + b

(x+2) = 0

x = -2

And,

(2 x-1) = 0

2 x = 1

X = 1/2

P(X) = 0

P(-2) = 0

2 × (-2)³ + a × (-2)² + (-2) + b = 0

2 × -8 + a × 4 - 2 + b = 0

-16 + 4 a - 2 + b = 0

4 a + b = 18 ---------(1)

Also X = 1/2

P(X) = 0

P(1/2) = 0

2 × (1/2)³ + a × (1/2)² + 1/2 + b = 0

2 × 1/8 + a × 1/4 + 1/2 + b = 0

1/4 + a/4 + 1/2 + b = 0

1 + a + 2 + 4 b /4 = 0

a + 4 b + 3 = 0

a + 4 b = -3 -----------(2)

From equation (1) and (2) we get,

(4 a + b = 18) × 1

(a + 4 b = -3) × 4

4 a + b = 18

4 a + 16 b = -12

Subtracting (2) from (1),

-15 b = 30

∴ b = -2

Putting value of b in (1),

4 a + (-2) = 18

∴ a = 5