Answer:
f(z) = (z -1)(z -10)(z -12)
Explanation:
For factoring polynomials of higher degree, a graphing calculator can provide information about real zeros. (see attached) A graph shows us the function has zeros at z=1, z=10, z=12. For a zero p, (z-p) is a factor. That means the factors are ...
f(z) = (z -1)(z -10)(z -12)
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If you notice that the sum of coefficients of the terms in the polynomial is zero, then you realize that z=1 is a zero. Any of various means can be used to factor out z-1 from the polynomial. Perhaps simplest is synthetic division. The table for that is shown in the second attachment. It tells you the factorization is ...
f(x) = (z -1)(z^2 -22z +120)
The quadratic is factored in the usual way: look for factors of 120 that have a sum of -22.
120 = (-1)(-120) = (-2)(-60) = (-3)(-40) = (-4)(-30) = (-5)(-24)
= (-6)(-20) = (-8)(-15) = (-10)(-12)
The factors -10 and -12 have a sum of -22, so the remaining binomial factors are (z -10)(z -12).
The factored form is ...
f(z) = (z -1)(z -10)(z -12)