🔶 Factoring Polynomials
Factoring is like reverse multiplication! Learn to break down polynomials into simpler parts.
🔍 What is Factoring?
Factoring means writing a polynomial as a product of simpler polynomials.
Why Factor?
- Simplifies expressions
- Helps solve equations
- Reveals important properties
Example: $x^2 + 5x + 6 = (x + 2)(x + 3)$
🎯 Greatest Common Factor (GCF)
Always look for the GCF first!
Example 1
Factor: $6x^3 + 9x^2 - 12x$
GCF: $3x$
$$3x(2x^2 + 3x - 4)$$
Example 2
Factor: $4x^2y + 8xy^2$
GCF: $4xy$
$$4xy(x + 2y)$$
🎨 Factoring Trinomials
Factor $x^2 + bx + c$ by finding two numbers that multiply to $c$ and add to $b$!
Example: $x^2 + 7x + 12$
Find two numbers that:
- Multiply to 12
- Add to 7
Numbers: 3 and 4 (since $3 \times 4 = 12$ and $3 + 4 = 7$)
$$x^2 + 7x + 12 = (x + 3)(x + 4)$$
⭐ Difference of Squares
$a^2 - b^2 = (a + b)(a - b)$
Examples
$x^2 - 16 = (x + 4)(x - 4)$
$9x^2 - 25 = (3x + 5)(3x - 5)$
$4x^2 - 49 = (2x + 7)(2x - 7)$
🔧 Factoring by Grouping
For four-term polynomials, group and factor!
Example: $x^3 + 3x^2 + 2x + 6$
Step 1: Group
$(x^3 + 3x^2) + (2x + 6)$
Step 2: Factor each group
$x^2(x + 3) + 2(x + 3)$
Step 3: Factor out common binomial
$$(x + 3)(x^2 + 2)$$
🌟 Real-World Applications
- 🎯 Problem Solving: Simplify complex algebraic fractions
- 📊 Graphing: Find x-intercepts of parabolas
- 🔬 Science: Analyze projectile motion
- 💼 Economics: Break-even point analysis
🎯 Practice Questions
Master these concepts with practice!
🔥 Challenge Questions
Ready for a challenge?