Standard Form of Polynomial
In this lesson, we'll learn how to write polynomials in their standard form — a neat, organized way that makes polynomials easy to work with and compare.
📚 Like Terms
Definition: Like terms are terms that have identical variable parts (same variables raised to the same powers).
Examples of Like Terms:
- 5x² and −3x² Like terms ✓
- 7xy and 2xy Like terms ✓
- 7 and −2 Like terms ✓ (constants)
- 5x² and 5x³ NOT like terms ✗
- 3xy and 3x²y NOT like terms ✗
Remember: Constants (numbers without variables) are like terms to each other!
🔧 Combining Like Terms
Definition: Combining like terms means replacing a sum of like terms with a single monomial.
Mathematical Basis:
This process uses three important properties:
- Commutative property: a + b = b + a
- Associative property: (a + b) + c = a + (b + c)
- Distributive property: ka + ma = (k + m)a
Example:
5x² − 3x² = (5 − 3)x² = 2x²
⭐ Definition: Standard Form of a Polynomial
A polynomial is in standard form when:
- Each term is a monomial in standard form
- There are no like terms (all like terms have been combined)
- Terms are arranged in decreasing order of degree
✓ Standard Form
7x² + 8xy − 5
Each term simplified, no like terms, ordered by degree✗ NOT Standard Form
3x² + 5 − 2x² + x
Contains like terms (3x² and −2x²)