Mathematics
The polynomial 2x3 - 7x2 + ax - 6 and x3 - 8x2 + (2a + 1)x - 16 leave the same remainder when divided by x - 2. Find the value of 'a'.
Answer
Given,
2x3 - 7x2 + ax - 6 and x3 - 8x2 + (2a + 1)x - 16 leave the same remainder when divided by x - 2.
x - 2 = 0 ⇒ x = 2
∴ On substituting x = 2 in 2x3 - 7x2 + ax - 6 and x3 - 8x2 + (2a + 1)x - 16 the values are equal.
∴ 2(2)3 - 7(2)2 + a(2) - 6 = (2)3 - 8(2)2 + (2a + 1)(2) - 16
⇒ 2(8) - 7(4) + 2a - 6 = 8 - 32 + 4a + 2 - 16
⇒ 16 - 28 + 2a - 6 = 8 - 32 + 4a + 2 - 16
⇒ 2a - 18 = 4a - 38
⇒ 4a - 2a = 38 - 18
⇒ 2a = 20
⇒ a = 10.
Hence, a = 10.
Related Questions
What number should be added to 3x3 - 5x2 + 6x so that when resulting polynomial is divided by x - 3, the remainder is 8 ?
What number should be subtracted from x3 + 3x2 - 8x + 14 so that on dividing it by x - 2, the remainder is 10 ?
If (x - 2) is a factor of the expression 2x3 + ax2 + bx - 14 and when the expression is divided by (x - 3), it leaves a remainder 52, find the values of a and b.
For the polynomial x5 - x4 + x3 - 8x2 + 6x + 15, the maximum number of linear factors is :
9
6
7
5