3.1 Introduction to Algebra
What is Algebra?
Algebra is the branch of mathematics that uses letters (called variables) to represent unknown numbers. It helps us solve problems where some quantities are unknown by creating equations and finding patterns.
If 5 chocolates cost ₹75, how much does one chocolate cost?
Let cost of one chocolate = x
Then: 5 × x = 75
So: x = 75 ÷ 5 = 15
Each chocolate costs ₹15!
Algebra Magic Trick
Think of a number. Add 5, multiply by 2, subtract 4, divide by 2, and subtract your original number.
The answer is always 3! Try it with different numbers.
Let your number be x:
Step 1: x + 5
Step 2: (x + 5) × 2 = 2x + 10
Step 3: 2x + 10 - 4 = 2x + 6
Step 4: (2x + 6) ÷ 2 = x + 3
Step 5: x + 3 - x = 3
The x always cancels out, leaving 3!
3.2 Polynomials
Understanding Polynomials
A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication (but not division by a variable).
Linear
Degree: 1
Graph: Straight line
Quadratic
Degree: 2
Graph: Parabola
Cubic
Degree: 3
Graph: S-shaped curve
Polynomial Builder & Grapher
Create your own polynomial and see its graph:
3.3 Remainder Theorem
The Remainder Theorem
When a polynomial f(x) is divided by (x - a), the remainder is equal to f(a).
Find the remainder when f(x) = x³ - 4x² + 2x - 3 is divided by (x - 2).
Using the Remainder Theorem:
f(2) = (2)³ - 4(2)² + 2(2) - 3
= 8 - 16 + 4 - 3 = -7
The remainder is -7.
Remainder Calculator
Enter a polynomial and a divisor to find the remainder:
3.4 Algebraic Identities
Key Algebraic Identities
These formulas are always true, no matter what values you substitute for the variables.
Square of Sum
Square of Difference
Difference of Squares
Identity Verifier
Choose values for a and b to verify the identities:
3.5 Factorisation
What is Factorisation?
Factorisation is the process of breaking down an expression into simpler parts (factors) that when multiplied together give the original expression.
Factorise: x² + 5x + 6
We look for two numbers that multiply to 6 and add to 5:
2 × 3 = 6 and 2 + 3 = 5
So: x² + 5x + 6 = (x + 2)(x + 3)
Factorisation Visualizer
Enter a quadratic expression to see its factors:
3.6 Division of Polynomials
Polynomial Long Division
Polynomial division works similarly to numerical long division. We divide the highest degree terms and subtract until the remainder has a lower degree than the divisor.
Divide (x³ - 2x² - 4) by (x - 3)
1. Divide x³ by x to get x²
2. Multiply (x - 3) by x² and subtract
3. Bring down next term and repeat
Result: x² + x + 3 with remainder 5
Polynomial Division Calculator
Enter two polynomials to perform division:
3.7 Greatest Common Divisor (GCD)
Finding GCD of Polynomials
The GCD of two polynomials is the highest degree polynomial that divides both of them without leaving a remainder.
Find GCD of (x² - 4) and (x² - 5x + 6)
Factor both polynomials:
x² - 4 = (x + 2)(x - 2)
x² - 5x + 6 = (x - 2)(x - 3)
Common factor: (x - 2)
GCD is (x - 2)
GCD Finder
Enter two polynomials to find their GCD:
3.8 Linear Equations in Two Variables
Solving Linear Systems
A linear equation in two variables represents a straight line. A system of two equations represents the point(s) where the lines intersect.
Solve the system:
Solution: x = 3, y = 2
Linear Equation Solver
Enter coefficients for two equations: