For what value of c will the polynomial P(x) = -2x^3 + cx^2 - 5x + 2 have the same remainder when it is divided by x - 2 and by x + 1?
We can use some polynomial "long division" to help us with this one.....
-2x^2 + [c- 4]x + [2c - 13]
----------------------------------------------------------------
x - 2 -2x^3 + cx^2 - 5x + 2
-2x^3 + 4x^2
----------------------------------------------------------------
(c -4)x^2 -5x
(c - 4)x^2 -2(c - 4)x
----------------------------------------------------------------
[2c - 13]x + 2
[2c - 13]x -2[2c - 13]
-----------------------------------------------------------------------
4c - 24
-2x^2 + [c + 2]x - [ 7 + c]
-------------------------------------------------------------
x + 1 -2x^3 + cx^2 - 5x + 2
-2x^3 - 2x^2
-----------------------------------------------------------------------
[c + 2)x^2 -5x
[c + 2]x^2 + [c + 2)x
------------------------------------------------------------------------
- [7 + c] x + 2
- [7 + c] x - [7 + c]
--------------------------------------------------------------------------
9 + c
And it's obvious that the remainders will be equal when.....
4c - 24 = 9 + c subtract c from both sides and add 24 to both sides
3c = 33 ..... so .......
c = 11
We can use some polynomial "long division" to help us with this one.....
-2x^2 + [c- 4]x + [2c - 13]
----------------------------------------------------------------
x - 2 -2x^3 + cx^2 - 5x + 2
-2x^3 + 4x^2
----------------------------------------------------------------
(c -4)x^2 -5x
(c - 4)x^2 -2(c - 4)x
----------------------------------------------------------------
[2c - 13]x + 2
[2c - 13]x -2[2c - 13]
-----------------------------------------------------------------------
4c - 24
-2x^2 + [c + 2]x - [ 7 + c]
-------------------------------------------------------------
x + 1 -2x^3 + cx^2 - 5x + 2
-2x^3 - 2x^2
-----------------------------------------------------------------------
[c + 2)x^2 -5x
[c + 2]x^2 + [c + 2)x
------------------------------------------------------------------------
- [7 + c] x + 2
- [7 + c] x - [7 + c]
--------------------------------------------------------------------------
9 + c
And it's obvious that the remainders will be equal when.....
4c - 24 = 9 + c subtract c from both sides and add 24 to both sides
3c = 33 ..... so .......
c = 11
That was quite a feat Chris,
I will do it using remainder theorem.
For what value of c will the polynomial P(x) = -2x^3 + cx^2 - 5x + 2 have the same remainder when it is divided by x - 2 and by x + 1?
When P(x) is divided by x-2 the remainder will be p(2)=-2*8+c*4-10+2 = -16+4c-8 = 4c-24
When P(x) is divided by x+1 the remainder will be p(-1)=-2*-1+c*1+5+2=2+c+7 = 9+c
If these are the same then
4c-24=9+c
3c=33
c=11