Determine the remainder when P(x)=x^5-2x^3+3x-3 is divided by x^3-x using the Remainder Theorem.Do not use long division.how to determine the...

P(x) = x^5-2x^3+3x-3.

To find
the remainder when divided by x^3-x.

x^3-x = x(x^2-1) =
(x)(x-1)(x+1).

So we can pressume that the remainder is  of
the form ax^2+bx+c a second degree expression ,1degree less than x^3-x. and the quotient
is Q(x).

Therefore
,

 P(x)=x^5-2x^3+3x-3 =  x(x+1)(x)(x-1)Q(x) + ax^2+bx+c.
..(1)

Put x= 0 in both sides in eq
(1):

 Put x = -1:

(-1)^5
- 2(-1)^3 +3(-3) -3 = (-1+1)(-1)(-1-1)Q(-1)
+a(-1)^2+b(-1)+c

-5+2-3-3 =
0+a-b+c.

-9 = a-b+c

a-b+c =
-9.......................(2)

Put x=
0

-3 = 0 +0+0+c. So c =
-3.

put x= 1:

1-2+3-3 =
0+a+b+c

-1 = a+b+c

a+b+c =
-1.............(3)

a-b+c = -
9.............(2)

Adding (1) and(2) , we get: 2a+2c = -10.
So a = -10-2c = -10 -2(-3) = -4. So a = -4/2 = -2.

Puttin
a= -2 and c = -3 in (3), we get: -2+b-3 = -1. So b = -1+5 =
4..

Therefore a=-2, b = 4 and
c=-3.

Therefore the assumed remainder ax^2+bx+c is
-2x^2+4x-3.