Question

Factor algebraic equations quick

Answer 1

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**:
1. Find two numbers that multiply to \( ac \) (product) and add to \( b \) (sum).
2. 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
1. Factor \( x^2 - 9 \):
\[
x^2 - 9 = (x - 3)(x + 3) \quad \text{(Difference of Squares)}
\]

2. Factor \( x^2 + 7x + 10 \):
\[
x^2 + 7x + 10 = (x + 5)(x + 2) \quad \text{(Product-Sum method)}
\]

3. 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!