Question 468064
here's a good reference on remainder theorem.

<a href = "http://www.mathsisfun.com/algebra/polynomials-remainder-factor.html" target = "_blank">http://www.mathsisfun.com/algebra/polynomials-remainder-factor.html</a>

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:

{{{graph(600,600,-10,10,-200,200,x^3 + 3x^2 - 8x + 10,170)}}}

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.