Mathematics
Given a polynomial f(x) = 2x3 - 7x2 - 5x + 4
Assertion (A): (x - 1) is not factor of f(x).
Reason (R): f(1) = -6.
Assertion (A) is true, but Reason (R) is false.
Assertion (A) is false, but Reason (R) is true.
Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).
Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).
Answer
Given a polynomial f(x) = 2x3 - 7x2 - 5x + 4
By the Factor Theorem, (x - r) is a factor of f(x) if and only if f(r) = 0.
(x - 1) is a factor of f(x) if and only if f(1) = 0.
f(1) = 2 . (1)3 - 7 . (1)2 - 5 . (1) + 4
= 2 - 7 - 5 + 4 = -6.
Thus, (x - 1) is not a factor of f(x).
So, Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).
Hence, option 3 is the correct option.
Related Questions
A polynomial in 'x' is divided by (x - a) and for (x - a) to be a factor of this polynomial, the remainder should be :
-a
0
a
2a
Given f(x) = ax2 + bx + c, a ≠ 0, b, c ∈ R.
Assertion (A): The factorisation of ax2 + bx + c is possible only if its discriminant = b2 - 4ac < 0.
Reason (R): To factorise ax2 + bx + c, split the coefficient of x into two real numbers such that their algebraic sum is b and their product is ac.
Assertion (A) is true, but Reason (R) is false.
Assertion (A) is false, but Reason (R) is true.
Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).
Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).
Find the remainder when 2x3 - 3x2 + 4x + 7 is divided by
(i) x - 2
(ii) x + 3
(iii) 2x + 1