Question

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. If (x + 1) and (x + 2) are the factors of x³ + px² + qx − 2 then p = 0.
2. 2x³ + x² + 17x + 46 is exactly divisible by x + 2.

Options

• Only (i)

• Only (ii)

• Both (i) and (ii)

• Neither (i) nor (ii)

Answer 1

Only (ii)

Explanation:

i.

Let f(x) = x³ + px² + qx − 2

∵ (x + 1) is a factor of f(x)

∴ f (−1) = 0

⇒ −1³ + p(−1)² + q(−1) − 2 = 0

⇒ −1 + p − q − 2 = 0

⇒ p − q − 3 = 0

⇒ p − q = 3   ...(i)

(x + 2) is a factor of f(x)

∴ f (−2) = 0

⇒ −2³ + p(−2)² + q(−2) − 2 = 0

⇒ −8 + 4p − 2q − 2 = 0

⇒ 4p − 2q − 10 = 0

Divide by 2:

2p − q = 5   ...(ii)

Subtract (i) from (ii),

(p − q) − (2p − q) = 5 − 3

2p − q − p + q = 2

p = 2

Therefore, statement (i) is invalid.

ii.

Let f(x) = 2x³ + x² + 17x + 46

Let x + 2 = 0

x = −2

F(−2) = 2(−2)³ + (−2)² + 17(−2) + 46

= 2(−8) + 4 − 34 + 46

= −16 + 4 − 34 + 46

= −12 − 34 + 46

= −46 + 46

= 0

Therefore, statement (ii) is valid.