The zeros (or roots) of a polynomial are the values of the variable that make the polynomial have a value of zero. If "z" is a zero of a polynomial then (x-z) will be a factor of the polynomial (and vice versa). So we can find zeros if we can factor the polynomial.
The greatest common factor of g(x) is 1 (which we rarely bother factoring out). g(x) has too many terms for the factoring patterns or for trinomial factoring. Since I do not see how to factor g(x) by grouping we are left with factoring by trial and error of the possible rational roots.
The possible rational roots of a polynomial are all the ratios, positive and negative, which can be formed using a factor of the constant term on top and a factor of the leading coefficient on the bottom. g(x)'s constant term is 26 (whose factors are 1, 2, 13 and 26). The leading coefficient is 1 (whose factors are 1's). So the possible rational roots of g(x) are:
+1/1, +2/1, +13/1, +26/1
which simplify to:
+1, +2, +13, +26
We can check to see if 1 or -1 are roots by mental math (since powers of them are easy). Neither of these turn out to be roots. If you know Descartes' Rule of Signs, then you will know that there can be no positive roots. (If you don't know this rule then you can just try out the positive possible roots and see for yourself.) So I am going to try -2 next. For numbers other than 1 and -1, it is usually easier to check a possible root by using synthetic division:
-2 | 1 6 21 26
---- -2 -8 -26
-----------------
1 4 13 0
The remainder is in the lower right corner. A zero remainder means that (x-(-2)) divided evenly into g(x). This means that -2 is a root. Also, the rest of the bottom row tells us what the other factor is. The "1 4 13" translates into . So g(x), partially factored, is:
or
The second factor is a quadratic. We can try factoring by patterns or trinomial factoring on it. Neither of these work. So will not factor further. But since it is a quadratic we can use the quadratic formula to find the remaining roots:
Simplifying...
At this point we should notice that there is a negative number in the square root. This means the remaining roots are going to be complex numbers. Since the problem says "find all roots" and not "find all real roots", I'm going to assume that we are supposed to find complex roots, too. Continuing to simplify:
x = -2 +- 3i
which is short for:
x = -2+3i or x = -2-3i
In standard a + bi form, the second would be -2 + (-3i). So the three roots/zeros of g(x) are -2, -2 + 3i and -2 + (-3i)