What is a polynomial? (We explain on train carriages)
The word “polynomial” sounds to many schoolchildren as something scary and difficult. But in fact, everything is very simple. Imagine a long freight train. It consists of separate trailers that are connected to each other. In mathematics, everything works exactly the same!
A Polynomial (or polynomial) is a mathematical expression that consists of several small pieces (monomials) connected by plus (➕) or minus (➖) signs.
What parts does it consist of? (We take it apart into screws)
Let's take a typical polynomial, which is often found in textbooks on algebra, and analyze it in parts: 3x2 − 4x + 7.
This expression is similar to a three-car train. Each such “car” of mathematicians is called amember. Here's what's hidden inside them:
- 🔢 Coefficients: These are ordinary numbers that stand in front of the letters. In our example, these are the numbers 3 and -4. They show how many times we took the letter.
- 🔤 Variables: These are letters (usually x, y, a or b). You can substitute any number instead.
- 🔝 Degrees: These are small numbers at the top right of the letter (for example, two in x2). They show how many times a letter is multiplied by itself.
- 🧍♂️ Free member: This is the single number at the very end of the chain (in our case it is +7). He does not have a partner letter, he walks on his own.
What types of polynomials are there?
Depending on how many “cars” there are in our mathematical train, they get different funny names:
| Title | How many parts? | Simple example |
|---|---|---|
| Monomial | 1 part (one car only) | 5x3 |
| Binomial | 2 parts (two cars) | 2x + 5 |
| Trinomial | 3 parts (three cars) | x2 − 4x + 4 |
| Polynomial | 4 or more parts | x4 + 3x3 − x2 + 8 |
🔥 A common question on a test: How to determine the degree of everything polynomial?
The degree of a polynomial is very easy to determine! You need to look at all the letters and find the one with the largest number on top (the highest degree).
For example, in the expression 2x5 + 3x2 − 7 the largest number at the top is five. So this is a polynomial ofthe fifth degree. It's simple: whoever is the oldest in the carriages is in charge!
Why are they needed in real life and IT?
It seems that these letters and numbers are only needed to torment schoolchildren in exams. But in fact, without polynomials, our modern world simply could not work!
- 🎮 Game Development:Have you played Angry Birds or any archery game? When a bird or arrow flies in an arc, the computer calculates its ideal flight path using a polynomial (a square trinomial that draws a parabola).
- 🎢 Architecture and design: The shape of roller coasters, the curves of modern buildings, car bodies and even the design of your smartphone are drawn in special programs using splines - complex polynomials that make lines smooth.
- 🔐 Safety on the Internet: When you enter password from your mail or transfer money from a bank card, your data are encrypted with the most complex cryptography algorithms. And many of them are based on polynomials with huge numbers!
- 📈 Economics: Using these formulas, economists and smart calculators predict how prices will rise, how much profit a store will bring, or how exchange rates will change.