Factoring algebraic equations can vary in complexity depending on the type of equation. Here are some common types and methods to factor them quickly:
1. Common Factor
For expressions like \( ax + ay \):
- Factor out the common term: \[ ax + ay = a(x + y) \]
2. Difference of Squares
For expressions like \( a^2 - b^2 \):
- Use the formula: \[ a^2 - b^2 = (a - b)(a + b) \]
3. Quadratic Trinomials
For a quadratic expression like \( ax^2 + bx + c \):
- Factor using the product-sum method:
- Find two numbers that multiply to \( ac \) (product) and add to \( b \) (sum).
- Rewrite and factor by grouping.
Example: \( x^2 + 5x + 6 \)
- Factors of 6 that add to 5 are 2 and 3.
- Factor: \[ x^2 + 5x + 6 = (x + 2)(x + 3) \]
4. Perfect Square Trinomials
For expressions like \( a^2 + 2ab + b^2 \) or \( a^2 - 2ab + b^2 \):
- Use the formulas: \[ a^2 + 2ab + b^2 = (a + b)^2 \] \[ a^2 - 2ab + b^2 = (a - b)^2 \]
5. Grouping
For expressions with four terms, like \( ax + ay + bx + by \):
- Group and factor: \[ (ax + ay) + (bx + by) = a(x + y) + b(x + y) = (a + b)(x + y) \]
6. Sum/Difference of Cubes
For expressions like \( a^3 + b^3 \) or \( a^3 - b^3 \):
- Use the formulas: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \] \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Example Problems
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Factor \( x^2 - 9 \): \[ x^2 - 9 = (x - 3)(x + 3) \quad \text{(Difference of Squares)} \]
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Factor \( x^2 + 7x + 10 \): \[ x^2 + 7x + 10 = (x + 5)(x + 2) \quad \text{(Product-Sum method)} \]
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Factor \( 3x^3 - 12x \): \[ 3x^3 - 12x = 3x(x^2 - 4) = 3x(x - 2)(x + 2) \quad \text{(Factor out common term, then Difference of Squares)} \]
These methods should help you factor algebraic expressions efficiently. If you have a specific expression in mind, feel free to share, and I can help you factor it!
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