Mathematics
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.
Answer
Both A and R are true and R is correct reason for A.
Reason
By factor theorem,
(x - a) is a factor of the polynomial f(x), if the remainder i.e. f(a) = 0.
Let f(x) = x36 - 3x35
When, x - 3 is a factor of f(x), then f(3) = 0, by factor theorem.
∴ Assertion (A) is true.
When, p(x) = x36 - ax35 is divided by x - a, we get :
Remainder, p(a) = a36 - a.a35
= a36 - a36
= 0.
Since, p(a) = 0, thus (x - a) is factor of p(x).
∴ Reason (R) is true.
Thus, Both A and R are true and R is correct reason for R.
Hence, option 3 is the correct option.
Related Questions
4x2 - kx + 5 leaves a remainder 2 when divided by x - 1. The value of k is :
-6
6
7
-7
If mx2 - nx + 8 has x - 2 as a factor, then :
2m - n = 4
2m + n = 4
2n + m = 4
n - 2m = 4
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.
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.