Understanding the Wavy Curve Method for Solving Polynomial Inequalities

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The Wavy Curve Method is a technique for solving polynomial inequalities of the form P(x)/Q(y) ≤ 0 and P(x)/Q(y) ≥ 0, where P(x) and Q(y) are polynomials. It involves sketching the graphs of P and Q on separate axes to identify intervals where the ratios are negative or positive. Understanding the roots and their multiplicities is crucial; roots with even multiplicity cause the curve to bounce off the axis, while those with odd multiplicity allow the curve to pass through. This method relies on basic polynomial sketching skills learned in calculus, focusing on root behavior rather than extreme values. The Wavy Curve Method effectively combines graphical analysis to solve polynomial inequalities.
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It is said to be a method to solve inequalities in the form of
\frac{P(x)}{Q(y)} \leq 0

\frac{P(x)}{Q(y)} \geq 0

P(x) and Q(y) are polynomials.
 
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It's basically to roughly sketch P and Q as graphs on two separate axis, from which we can observe which intervals they are negative or positive. We can combine the information of both to determine when the ratios are negative or positive.

You should have learned how to roughly sketch polynomials in calculus. For this purpose, you only need to find the roots and how the graph passes through them (bounce back off, or pass through the axis), and not the values of extreme values or where exactly they occur. The quick rule is that if the root has even multiplicity, the curve bounces back off the axis, while roots with odd multiplicity pass straight through. You can use Calculus to check that if you want.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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