Polynomials 3x^3- 5x^2 +kx -2 and -x^3-x^2+7x + k leave the same remainder when divided by x+2. Find the value of k.
Answers
Answer:
The value of k is 12
Step-by-step explanation:
Given polynomials are:
3x³ - 5x² + kx - 2
-x³ - x² + 7x + k
They are divided by (x + 2)
It is given that they leave the same remainder when divided.
Let the remainder be 'n'
Therefore, 3x³ - 5x² + kx - 2 + n will be completely divisible by x + 2
=> x = -2 will satisfy the given polynomial.
=> 3(-2)³ - 5(-2)² + k(-2) - 2 + n = 0
=> 3 (-8) - 5(4) + k(-2) - 2 + n = 0
=> - 24 - 20 - 2k - 2 + n = 0
=> - 46 - 2k + n = 0
=> n = 46 + 2k --(i)
Similarly,
x = -2 will satisfy -x³ - x² + 7x + k + n
=> -(-2)³ - (-2)² + 7(-2) + k + n = 0
=> - (-8) - (4) + 7(-2) + k + n = 0
=> 8 - 4 - 14 + k + n = 0
=> -10 + k + n = 0
=> n = 10 - k --(ii)
On equating equation (i) and (ii), we get,
46 + 2k = 10 - k
=> 2k + k = 10 - 46
=> 3k = - 36
=> k = - 12
Therefore, the value of k is 12