The polynomial $$g(x) = x^3 - x^2 - (m^2 + m) x + 2m^2 + 4m + 2$$is divisible by $x-4$ and all of its zeroes are integers. find all possible values of $m$.

m=5

Step-by-step explanation:

In order to solve this problem we need to start by finding the possible m-values for which x=4 is a valid zero for the polynomial. We can do so by doing a synthetic division. Our coefficients are 1, -1, and respectively. So our long division looks like the one in the picture. (See attached picture).

A you may see in the division, we will get a remainder of .

For x=4 to be a valid answer, the remainder must be equal to zero, so we need to set that remainder equal to zero and solve for m, so we get:

when solving for m we get that:

this means that either m=5 and m=-5.

These are the possible values for m, but we need to check if they will both give us a positive answer when finding the zeros for our function.

When doing the long division, we got a factored function of:

so we can substitute both m=5 and m=-5 into the quadratic part of our function and solve for x so we get two equations:

test for m=5:

which can be solved by factoring so we get:

(x+6)(x-3)=0 which yields two integers as answers:

x=-6 and x=3

so m=5 is a valid answer. Let's test for m=-5 now.

this one can only be solved by using the quadratic formula, so we get:

when inputing the corresponding values we get that:

which yields:

since 41 is not a perfect square, this formula would give us decimal answers so m=-5 is not a valid value for m, since it won't give you integer zeros.

You can prove the answer by finding the possible zeros of the final function which is:

which has the following zeros:

x=-6, x=3 and x=4

So the only possible value for m is m=5

no pictures of the graph