Comprehensive Study Notes on Factoring Polynomials

Factoring Basics

Introduction to Factoring

• Factoring is the process of splitting an expression into products of simpler factors.
• The main focus is on understanding the Greatest Common Factor (GCF).
• GCF Definition: The largest factor that two or more numbers share.

Understanding Factors

• Factors: Numbers or variables that can be multiplied together to get a term.
• Key Point: Factors are involved in multiplication, not addition.
  • Example: For the term 15, factors include 3 and 5 because $3 imes 5 = 15$.
  • Any expression can be factored if you can split it into components multiplied together.

Types of Factors

1. Individual Number Factors

• For example, for 15:
  • Factors include: 1, 3, 5, and 15. Pairs of factors: (1, 15) and (3, 5).
• Composite numbers have multiple factors while prime numbers have only two: 1 and itself.

2. Variable Factors

• Example with a variable: If we have $x^7$,
  • Possible factors include: $x, x^2, x^3, x^4, x^5, x^6, x^7$.

Common Factors

• Common Factors: Shared factors between numbers or expressions.
• Example: For 12 and 28:
  • Factors of 12: 1, 2, 3, 4, 6, 12
  • Factors of 28: 1, 2, 4, 7, 14, 28
  • Common Factors: 1, 2, 4

Greatest Common Factor (GCF)

• GCF: The largest of the common factors.
• Practice finding GCFs through a systematic approach:
  • Method 1: List all factors and identify the highest.
  • Method 2: Use prime factorization to identify GCF.
  • For instance, 45 and 75:
    • Prime factorization: 45 = $3^2 imes 5$, 75 = $3 imes 5^2$.
    • GCF is the product of the smallest powers: $3^1 imes 5^1 = 15$.

GCF with Variables

• When dealing with variables, look for the lowest exponent among the variable terms.
  • Example: GCF of $x^2, x^4, x^5$ is $x^2$.

Factoring Process

1. Identify the GCF: Examine all terms involved, including numbers and variables.
2. Write the GCF in Front: Factor it out and write it outside the parentheses.
3. Create Parentheses: Identify remaining terms after factoring.
4. Check Your Work: Ensure the factored form, when multiplied, returns to the original expression.

Examples of Factoring:

• Example: Factor $6x + 14$

  • gcf is 2.
  • $2(3x + 7)$

• Example: $-8y^2 - 63y + 27y^4$

  • GCF is -9y².
  • After factoring: $-9y²(-8y^4 + 13y + 3)$

Grouping Method for Four Terms

• Group pairs of terms to identify common factors, simplifying the equation.

  • For example: $x^{2}+7x+4−3y$ can be grouped and factored accordingly.

• After grouping, if the two grouped terms yield the same factor, factor that common factor out, creating a new pairs of parentheses for the remaining terms.

Practice Problems

1. Find the GCF of 12 and 28.
2. Factor the expression $x^2 + 7x + 12$.
3. Factor the expression $10xy + 15xz$.
4. Solve for GCF: $-8y^5 + 12y^3 - 4y^2$.