Finding value of polynomial using the remainder theorem

[frozenbananas]

Homework Statement

Find the indicated value of the polynomial using the Remainder Theorem
p(x)=2x^3-2x^2+11x-100; find p(3)

Homework Equations

p(x)=2x^3-2x^2+11x-100

The Attempt at a Solution

Synthetic division
3] 2 -2 11 -100
6 12 69
2 4 23 [-31
answer: p(3)=-31

im not sure because i just followed the example in the book and applied it to this problem but the one in the book had another polynomial it was divided by to find the remainder. please help me, i have a test in the morning and I am freaking out like crazy, thanks!

 

[HallsofIvy]

Yes, that is correct. Esssentially what you have shown is that
$$2x^3- 2x^2+ 11x- 100= (x- 3)(2x^2+ 4x+ 23)- 31$$

Putting x= 3 makes x- 3= 0 so whatever the number in the second paretheses is, the value is just -31.

Certainly, it wouldn't have been that hard for you to check it yourself by evaluating directly: $3^3= 27$ so $2(3^3)= 2(27)= 54$. $-2(3^2)=-2(9)= -18$. 11(3)= 33 so the whole thing is 54- 18+ 33- 100= 36+ 33- 100= 69- 100= -31.