Mathematics
Using the Remainder Theorem, find the remainders obtained when x3 + (kx + 8)x + k is divided by x + 1 and x - 2. Hence find k if the sum of the two remainders is 1.
Factorisation
5 Likes
Answer
By remainder theorem,
If f(x) is divided by (x - a), then remainder = f(a).
Let f(x) = x3 + (kx + 8)x + k = x3 + kx2 + 8x + k
Given,
Divisor :
⇒ x + 1 = 0
⇒ x = -1
On dividing x3 + kx2 + 8x + k by x + 1, we get :
⇒ f(-1) = (-1)3 + k(-1)2 + 8(-1) + k
= -1 + k - 8 + k
= 2k - 9.
Divisor :
⇒ x - 2 = 0
⇒ x = 2.
On dividing x3 + kx2 + 8x + k by x - 2, we get :
⇒ f(2) = (2)3 + k(2)2 + 8(2) + k
= 8 + 4k + 16 + k
= 5k + 24
Given,
Sum of two remainders is 1.
⇒ 2k - 9 + 5k + 24 = 1
⇒ 7k + 15 = 1
⇒ 7k = 1 - 15
⇒ k =
⇒ k = -2.
Hence, the value of k = -2.
Answered By
2 Likes
Related Questions
Find a if the two polynomials ax3 + 3x2 - 9 and 2x3 + 4x + a leave the same remainder when divided by (x + 3).
If (2x3 + ax2 + bx - 2) when divided by (2x - 3) and (x + 3) leaves remainders 7 and -20 respectively, find values of a and b.
Using factor theorem, show that:
(x - 3) is a factor of (x3 + x2 - 17x + 15).
Using factor theorem, show that:
(x + 1) is a factor of (x3 + 4x2 + 5x + 2).