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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.

Solving Identity Equations

         

How many real values of \(x\) satisfy the following equation \(9^x - 6 \cdot 3^x - 55 = 0\)?

If the equation \[3x+4y+5=ax+by+c=-dx-ey-f\] holds for all numbers \(x\) and \(y\), what is the value of \(cf-be-ad\)?

Given that \[ax+4y+cz+23=7x+by+4z+d\] is an algebraic identity in \(x\), \(y\) and \(z\), what is the value of \(a+b+c+d\)?

Given that \[ax^2+bx+c+7x^2+8x-12=0\] is an algebraic identity, what is the value of \(abc\)?

If the following holds for all real numbers \(x\) and \(y\), \[(ax+by+c)+(5x+5y-19)=0,\] what is the value of \(a+b+c\)?

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