Addition and Subtraction of Polynomials
In this lesson, we will learn how to add and subtract polynomials by applying bracket rules and combining like terms. These are fundamental skills in algebra that you'll use throughout your mathematical journey.
📗 Adding Polynomials
The Process:
- Write polynomials in parentheses with a plus sign between them
- Remove the parentheses (keeping all signs unchanged)
- Combine like terms
Rule for Opening Brackets with PLUS
When there's a plus sign before brackets:
→ Remove the brackets
→ Keep ALL signs unchanged
+(a + b − c) = a + b − c
+(3x² − 5x + 2) = 3x² − 5x + 2
Example:
(3x² + 2x) + (5x² − 4x)
= 3x² + 2x + 5x² − 4x
= 8x² − 2x
📕 Subtracting Polynomials
The Process:
- Write polynomials in parentheses with a minus sign between them
- Remove the parentheses (changing ALL signs in the second polynomial)
- Combine like terms
Rule for Opening Brackets with MINUS
When there's a minus sign before brackets:
→ Remove the brackets
→ Change ALL signs to their OPPOSITE
−(a + b − c) = −a − b + c
−(3x² − 5x + 2) = −3x² + 5x − 2
Example:
(3x² + 2x) − (5x² − 4x)
= 3x² + 2x − 5x² + 4x
= −2x² + 6x
🔍 Why Does the Sign Change Work?
The minus sign before brackets means we multiply each term inside by −1. This is the distributive property in action!
The Math Behind It:
- −1 × (a) = −a
- −1 × (−b) = +b
- −1 × (+c) = −c
Distributing the Negative:
−(3x² − 5x + 2)
= (−1)(3x²) + (−1)(−5x) + (−1)(2)
= −3x² + 5x − 2
Sign Change Visualization
Each term is multiplied by −1: