Question

The polynomial 3x³ + 8x² − 15x + k and one of its factors as (x − 1)

Assertion (A): The value of k = 4

Reason (R): x − 1 = 0 ⇒ x = 1
∴ 3(1)³ + 8(1)² − 15 × (1) + k = 0

पर्याय

• A is true, R is false.

• A is false, R is true.

• Both A and R are true and R is the correct reason for A.

• Both A and R are true and R is the incorrect reason for A.

Answer 1

Both A and R are true and R is the correct reason for A.

Explanation:

Let, f(x) = 3x³ + 8x² − 15x + k

By factor theorem,

(x − a) is a factor of the polynomial f(x), if the remainder i.e. f(a) = 0.

⇒ x − 1 = 0

⇒ x = 1.

Given,

x − 1 is one of the factors of f(x).

∴ f(1) = 0

⇒ 3.(1)³ + 8.(1)² − 15.1 + k = 0

⇒ 3.1 + 8.1 − 15 + k = 0

⇒ 3 + 8 − 15 + k = 0

⇒ −4 + k = 0

⇒ k = 4.

Thus, both A and R are true and R is correct reason for A.