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Polynomial Arithmetic

From profit/cost models to the flight of a baseball to approximating the motion of a wave, polynomials are a mathematical way to represent values which are sums of powers of variables.

Using the Perfect Square Identity

If \[x^2+20xy+100y^2=(x+ay)^2,\] what is the value of \(a\)?

If \[x^2+121-(x-a)^2=22x,\] what is the value of \(a\)?

If \[16x^2+56xy+49y^2=(ax+by)^2,\] what is the value of \(ab\)?

Given \[a-b=\sqrt{42+12\sqrt{6}}, b-c=\sqrt{42-12\sqrt{6}},\] what is the value of \[a^2+b^2+c^2-ab-bc-ca?\]

What is the coefficient of \(x\) in the expansion of \( (x+3 ) ^ 2 \)?

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