Lagrange Polynomials
Explicit interpolation polynomial.
Key idea
Explicit interpolation polynomial.
In depth
P(x) = Σ yᵢ Lᵢ(x), with Lᵢ(x) = Π_{j≠i}(x − xⱼ)/(xᵢ − xⱼ). Each Lᵢ equals 1 at xᵢ and 0 at the other nodes. A direct formula for interpolation.
Worked example
Through (0, 1), (1, 3): P(x) = 1·(x − 1)/(0 − 1) + 3·(x − 0)/(1 − 0) = 2x + 1.
Real-world uses
- Drawing a smooth curve through measured points in computer graphics and surveying
Common mistake
Dividing by zero if points repeat.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Lagrange Polynomials?
Explicit interpolation polynomial. It is a topic at the expert level in Differential Equations, Abstract Algebra & Applied Math.
Where is Lagrange Polynomials used in real life?
Real-world examples: Drawing a smooth curve through measured points in computer graphics and surveying.
What should I learn before Lagrange Polynomials?
Make sure you are comfortable with Interpolation, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with Lagrange Polynomials?
Dividing by zero if points repeat.
How can I practise Lagrange Polynomials?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Lagrange Polynomials?
Move on to Splines next.
Related expert topics in Differential Equations, Abstract Algebra & Applied Math
- Exact Equations – M dx + N dy = 0 with Mᵧ = Nₓ.
- Bernoulli Equations – Substitute v = y¹⁻ⁿ to linearise.
- Runge–Kutta Methods – Higher-accuracy numerical solvers.
- Second-Order Homogeneous Equations – Use the characteristic equation.
- Undetermined Coefficients – Guess the particular solution form.
- Variation of Parameters – General method for particular solutions.