Answer:
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, [tex]-(m^{2}+m)[/tex] and [tex](2m^{2}+4m+2)[/tex] 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 [tex]-2m^{2}+50[/tex].
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:
[tex]-2m^{2}+50=0[/tex]
when solving for m we get that:
[tex]2m^{2}=50[/tex]
[tex]m^{2}=25[/tex]
[tex]m=\pm 5[/tex]
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:
[tex]g(x)=(x-4)(x^{2}+3x+(-m^{2}-m+12))[/tex]
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:
[tex]x^{2}+3x+(-25-5+12)=0[/tex]
[tex]x^{2}+3x-18=0[/tex]
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.
[tex]x^{2}+3x+(-25+5+12)=0[/tex]
[tex]x^{2}+3x-8=0[/tex]
this one can only be solved by using the quadratic formula, so we get:
[tex]x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}[/tex]
when inputing the corresponding values we get that:
[tex]x=\frac{-3\pm \sqrt{(3)^{2}-4(1)(-8)}}{2(1)}[/tex]
which yields:
[tex]x=\frac{-3\pm \sqrt{41}}{2}[/tex]
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:
[tex]g(x)=x^{3}-x^{2}-30x+72[/tex]
which has the following zeros:
x=-6, x=3 and x=4
So the only possible value for m is m=5
