Question

If p(x)=2x^3+5x^2-3x-2 is divided by x-1 then find the remainder.

Answer 1

For the remainder theorem we generally use synthetic division.
If (x+2) is a factor of P(x) then x=-2 will be a zero of P(x)

we have P(x) = 2x2-3x+4
The coefficients in descending order from the variable with the highest
exponent to the constant are 2, -3, 4

Write the -2 on the left with a diagonal separating it from the coefficients:

-2 / 2 -3 4 1)bring down the first coefficient (2)
-4 14 2)multiply that by the -2 giving -4
-------------- 3)bring that product up under the next coefficient
2 -7 18 add down.
4 Repeat to the end. Last number is the remainder

The remainder is 18. The remainder theorem says that f(-2) = 18

The polynomial we get when we divide is
(2x2 -3x+4)/(x+2)= 2x-7+ 18/(x+2)