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.
