Question 468064: 4. Use the remainder theorem to:
(A)find the remainder when f(x) is divided by x – c, and
(B)determine whether x – c is a factor of f(x).
Please show all of your work.
f(x)=x³+3x²-8x+10;x-5
I appreciate any help, thanks.
Found 2 solutions by ewatrrr, Theo:
Answer by ewatrrr(24785)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! here's a good reference on remainder theorem.
http://www.mathsisfun.com/algebra/polynomials-remainder-factor.html
the gist of what's in the reference is as follows:
The remainder theorem states:
When you divide a polynomial f(x) by x-c the remainder r will be f(c)
The factor theorem states:
When f(c)=0 then x-c is a factor of the polynomial
what does this mean to your problem?
your equation is:
f(x) = x^3 + 3x^2 - 8x + 10
you want to divide this by (x-5)
you want to know whether (x-5) is a factor of x^3 + 3x^2 - 8x + 10
from the remainder theorem, if (x-5) is a factor of x^3 + 3x^2 - 8x + 10, then f(5) must be equal to 0.
replacing x with 5 in the equation, you get:
(5)^3 + 3(5^2) - 8(5) + 10 = 125 + 75 - 40 + 10 = 170
since the answer is not 0, this means that (x-5) is NOT a factor of x^3 + 3x^2 - 8x + 10.
being a factor of the equation, means that (x-5) would have been a root of the equation.
if (x-5) was a root of the equation, then the graph of the equation should have shown that the value of y when x = 5 was 0.
the graph of the equation is shown below:
from the graph, it looks like the graph might have a root at x = -5, but definitely NOT at x = 5.
if the equation had a root at x = -5, then (x+5) would be a factor of the equation, and when you divided the equation by (x+5), you should get a remainder of 0.
f(-5) = (-5)^3 + 3*(-5)^2 - 8*(-5) + 10 = -125 + 75 + 40 + 10 = 0
since the remainder is 0, this means that (x+5) IS a factor of the equation.
unfortunately, you were not asked that.
you were asked to find if (x-5) was a factor of the equation.
it is not.
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