Understanding the Wavy Curve Method for Solving Polynomial Inequalities

  • Thread starter Thread starter Smarty7
  • Start date Start date
  • Tags Tags
    Curve Method
AI Thread Summary
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.
Smarty7
Messages
22
Reaction score
0
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.
 
Mathematics news on Phys.org
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 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Thread 'Non-orthogonal bases'

Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
View full post »
Thread 'Imaginary Pythagoras'

Thread 'Imaginary Pythagoras'

I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...
View full post »

Similar threads

I A doubt about the multiplicity of polynomials in two variables
Replies
8
Views
2K
A Integral Points of an Elliptic Curve over a Cyclotomic Tower
Replies
5
Views
2K
I Partial Differential Equation solved using Products
Replies
2
Views
2K
MHB How to solve following optimization problem?
Replies
2
Views
1K
MHB Real Roots of Composite Polynomials: Solving P(Q(x))=0
Replies
1
Views
1K
I Rodrigues' Formula for Laguerre equation
Replies
3
Views
1K
MHB Solve Inequality & Simul Eqns: Alg Workings Q (a,b,c)
Replies
1
Views
1K
B Need assistance with an unusual quadratic solution method
Replies
4
Views
1K
MHB Real Number Pairs $(p,\,q)$ Satisfying Inequality
Replies
2
Views
1K
A Why the triangle inequality is greater than the 2 max{f(x),g(x)}
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top