Mathematics
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 x (1) + k = 0
options
A is true, R is false.
A is false, R is true.
Both A and R are true and R is correct reason for A.
Both A and R are true and R is incorrect reason for A.
Factorisation
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Answer
Both A and R are true and R is correct reason for A.
Reason
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.
Hence, Option 3 is the correct option.
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Related Questions
If mx2 - nx + 8 has x - 2 as a factor, then :
2m - n = 4
2m + n = 4
2n + m = 4
n - 2m = 4
Two polynomials x36 - 3x35 and x - 3.
Assertion (A) : If x - 3 is a factor of x36 - 3x35, the remainder is zero.
Reason (R) : The polynomial x - a is factor of polynomial p(x) = x36 - ax35, if p(a) = 0
options
A is true, R is false.
A is false, R is true.
Both A and R are true and R is correct reason for A.
Both A and R are true and R is incorrect reason for A.
The polynomial x2 + x + b has (x + 3) as a factor of it.
Statement 1: The value of b is -4.
Statement 2: (x + 3) is a factor of x2 + x + b ⇒ (3)2 + 3 + b = 0.
option
Both the statements are true.
Both the statements are false.
Statement 1 is true, and statement 2 is false.
Statement 1 is false, and statement 2 is true.
A polynomial x4 - 13x2 + 36.
Statement 1: x - 2 is a factor of x4 - 13x2 + 36.
Statement 2: (2)4 - 13 x (2)2 + 36 = 0.
option
Both the statements are true.
Both the statements are false.
Statement 1 is true, and statement 2 is false.
Statement 1 is false, and statement 2 is true.