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
(3x + 5) is a factor of the polynomial (a - 1)x3 + (a + 1)x2 - (2a + 1)x - 15. Find the value of 'a'. For this value of 'a', factorise the given polynomial completely.
Using remainder theorem, find the value of k if on dividing 2x3 + 3x2 - kx + 5 by (x - 2) leaves a remainder 7.
Find the value of 'a' if x - a is a factor of the polynomial 3x3 + x2 - ax - 81.
While factorizing a given polynomial, using remainder and factor theorem, a student finds that x + 3 is a factor of 2x3 - x2 - 5x - 2.
(a) Is the student's, solution correct stating that (x + 3) is a factor of the given polynomial?
(b) Give a valid reason for your answer.
(c) Factorize the given polynomial completely.