Mathematics
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.
Answer
Both the statements are true.
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) = x4 - 13x2 + 36
⇒ f(2) = 24 - 13 x 22 + 36
= 16 - 52 + 36
= 0
Since, f(2) = 0,
So, x - 2 is factor of x4 - 13x2 + 36.
∴ Statement 1 is correct.
Also,
⇒ 24 - 13 x 22 + 36 = 0.
∴ Statement 2 is correct.
Hence, option 1 is the correct option.
Related Questions
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.
When x3 + 3x2 - mx + 4 is divided by x - 2, the remainder is m + 3. Find the value of m.
What should be subtracted from 3x3 - 8x2 + 4x - 3, so that the resulting expression has x + 2 as a factor ?