Answer:
p(x) = x^12 -19x +84
Explanation:
Adding the given equations yields ...
2·alpha = 19+5
alpha = 24/2 = 12
Then beta = 19-12 = 7
The factored form of p(x) is then ...
p(x) = (x -12)(x -7)
Multiplying this out gives ...
p(x) = x^2 -19x +84
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Another way to get there is to realize that ...
p(x) = x^2 -(alpha+beta)x +(alpha·beta)
The constant term can be computed from the given sum and difference as ...
alpha·beta = ((alpha+beta)^2 -(alpha-beta)^2)/4 = (19^2 -5^2)/4 = 84
Then ...
p(x) = x^2 -19x +84