Answer: (x-4) is not a factor of the given polynomial.
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Explanation:
Rule: If (x-k) is a factor of p(x), then p(k) = 0. This is a special case of the remainder theorem. We can take this in reverse to say that if p(k) = 0, then (x-k) must be a factor of p(x).
In this case, we have k = 4 and we can find that:
p(x) = 2x^4-9x^3-20x^2+147x-180
p(4) = 2(4)^4-9(4)^3-20(4)^2+147(4)-180
p(4) = 24
The result we get is not zero, so that means (x-4) is not a factor of p(x).
