Polynomial Equation Find Values of "m" and "n"?

Question

The polynomial g(x) = mx^3 + nx^2 - x + 3 has a factor of (x+2). when g(x) is divided by (x-1) the remainder is 3. Determine values of "m" and "n"

So here is what I know.
Since g(-2) = 0 Therefore (x+2) is a factor. 
Since f(1) = 3 Therefore (x-1) equals remainder 3.

Equation 1: g(x) = m(-2)^3 + n(-2)^2 - (-2) + 3 = 0 
=> m(-8) + n(4) + 5 = 0 
=> m(-8) + n(4) = -5

Equation 2: g(x) = m(1)^3 + n(1)^2 - (1) + 3 = 3
=> m(1) + n(1) + 2 = 3
=> m(1) + n(1) = 1

Now in order to find "m" or "n" I must be able to subtract the two final equations, but I have made a mistake somewhere or am not using the right method in order to be able to simply subtract the two equations.
So can someone please help me.

larinon · Accepted Answer

You are right so far.  Looking at your two equations, we can write them:
-8m + 4n = -5
m + n = 1

If you want to solve this system using the elimination method, that means that when we add the equations together, one of the variables will cancel out.  In order to make that happen, we must first multiply one of the equations by a number.  Notice that we have a -8m in the first equation.  If we multiply the second equation by 8, that will give us 8m, so that when we add the -8m and the 8m, they will cancel out to 0.

So multiplying the second equation (all terms) by 8, we now have the following two equations:
-8m + 4n = -5
8m + 8n = 8

Now adding those together, we get:  12n = -3
From there we can solve for n by dividing by 12 --> n = -.25

After this, you can use the fact that m+n=1 from our second equation to find m.  You should get m = 1.25

Polynomial Equation Find Values of "m" and "n"?

1 Answer