If alpha and beta are the zero of a quadratic polynomial p(x) find p(x) if alpha + beta=19 and alpha-beta=5

p(x) = x^12 -19x +84

Step-by-step 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

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

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