**Factoring Techniques**

This section covers various methods for **factoring polynomials with different numbers of terms**, including:

- Factoring out the Greatest Common Factor (GCF)
- Difference of Squares
- Sum or Difference of Cubes
- Factoring Quadratics
- Factoring by Grouping
- Other Factoring Techniques

**Highlight**: Always factor out the Greatest Common Factor (GCF) first, if there is one.

**Example**: For difference of squares: a² - b² → (a+b)(a-b)

**Vocabulary**: Difference of Squares - A special case of factoring where a polynomial is in the form a² - b².

**Example**: For sum of cubes: a³ + b³ → (a+b)(a² - ab + b²)

**Example**: For difference of cubes: a³ - b³ → (a-b)(a² + ab + b²)

**Highlight**: When factoring quadratics (ax² + bx + c), find two factors of ac that add up to b.

**Example**: Factoring x² + 8x + 12:

- Factors of 12 that add up to 8 are 6 and 2
- Rewrite as x² + 6x + 2x + 12
- Factor by grouping: x(x+6) + 2(x+6)
- Final result: (x+2)(x+6)

**Vocabulary**: Factor by Grouping - A method used to factor polynomials with four or more terms by separating them into groups.

**Polynomial Long Division**

This section explains the process of **polynomial long division**, which is similar to regular long division but with polynomials.

**Highlight**: Don't forget to use placeholders for any missing degrees in the polynomial.

**Example**: Dividing (2x² + x - 3) by (x + 1):

- Divide first terms: 2x² ÷ x = 2x
- Multiply: 2x(x + 1) = 2x² + 2x
- Subtract: 2x² + x - 3 - (2x² + 2x) = -x - 3
- Repeat the process with -x - 3
- Final result: 2x - 1 with a remainder of -2

**Synthetic Division**

The guide concludes with an explanation of synthetic division, a shortcut method for dividing polynomials.

**Vocabulary**: Synthetic Division - A simplified method of polynomial long division, particularly useful when dividing by a linear factor.

**Example**: Synthetic division of 2x⁴ - 7x³ - 7x + 1 by (x - 4):

- Write coefficients: 2 -7 0 -7 1
- Use 4 as the divisor (root of x - 4)
- Bring down first coefficient: 2
- Multiply 2 by 4 and add to -7: 2(4) + (-7) = 1
- Continue this process for all terms
- Final result: 2x³ + x² + 4x + 9 with a remainder of 37

**Highlight**: Use zero as a placeholder for missing terms in the polynomial during synthetic division.