Consider the function stated below.
f(x)=x3+6∈Z7[x]
Make the observations as stated below.
f(1)=1+6=7=0(mod7)
Therefore, 1 is a zero of f(x).
Similarly;
f(2)=23+6=14=7×2=0(mod7)
Therefore, 2 is a zero of f(x)
Similarly;
f(4)=43+6=70=7×10=0(mod7)
Therefore, 4 is a zero of f(x).
Now, since {1,2,4} is a set of zeros of f(x);
Therefore, by virtue of the Factor Theorem, conclude that;{(x−1),(x−2),(x−4)} is a set of factors of f(x) .
This implies;
(x−1)(x−2)(x−4)∣x3+6
Also, note that;
deg((x−1)(x−2)(x−4))=3=deg(x3+6)
Hence, the required solution is x3+6=(x−1)(x−2)(x−4).
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