SOLUTION: I am trying to study for a test and I can not figure how to do this question. The polynomial f(x) divided x-3 results in a quotient of x^2+3x-5 with a remainder of 2. Find f(3).

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I am trying to study for a test and I can not figure how to do this question. The polynomial f(x) divided x-3 results in a quotient of x^2+3x-5 with a remainder of 2. Find f(3).      Log On


   

Question 78471: I am trying to study for a test and I can not figure how to do this question.
The polynomial f(x) divided x-3 results in a quotient of x^2+3x-5 with a remainder of 2. Find f(3).
a. -5
b. -2
c. 2
d. 3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
We know that
f%28x%29%2F%28x-3%29=x%5E2%2B3x-5%2B2%2F%28x-3%29 "The polynomial f(x) divided x-3 results in a quotient of x^2+3x-5 with a remainder of 2"
So multiply both sides by x-3 to solve for f(x)
cross%28x-3%29%28f%28x%29%2Fcross%28x-3%29%29=%28x-3%29%28x%5E2%2B3x-5%2B2%2F%28x-3%29%29

f%28x%29=%28x-3%29%28x%5E2%2B3x-5%2B2%2F%28x-3%29%29
Now let y=x-3 to make multiplication easier
f%28x%29=y%28x%5E2%2B3x-5%2B2%2Fy%29
f%28x%29=x%5E2y%2B3xy-5y%2B2%2Ay%2Fy%29 Distribute
f%28x%29=x%5E2y%2B3xy-5y%2B2%2Across%28y%29%2Fcross%28y%29%29 Notice the y's cancel on the last term
f%28x%29=x%5E2%28x-3%29%2B3x%28x-3%29-5%28x-3%29%2B2 Now replace y with x-3
f%28x%29=x%5E2%2Ax-x%5E3%2A3%2B3x%2Ax-3x%2A3-5%2Ax-5%2A3%2B2 Distribute
f%28x%29=x%5E3-3x%5E2%2B3x%5E2-9x-5x%2B15%2B2
f%28x%29=x%5E3-14x%2B17 Combine like terms

So our function f(x) is f%28x%29=x%5E3-14x%2B17. Notice if we divide f(x) by x-3 we get our original problem:
f%28x%29%2F%28x-3%29=%28x%5E3-14x%2B17%29%2F%28x-3%29=x%5E2%2B3x-5%2B2%2F%28x-3%29

Now since we know the function is f%28x%29=x%5E3-14x%2B17 then f(3) is

f%283%29=%283%29%5E3-14%283%29%2B17=27-42%2B17=2
So in other words, f(3)=2. So the answer is c)