7.1 Greatest Common Factor and Factor by Grouping

Factoring is a key skill in algebra, helping simplify complex expressions. The greatest common factor (GCF) is the largest factor shared by all terms in an expression. Finding the GCF allows us to break down polynomials into simpler forms.

Factoring by grouping is a technique for handling polynomials with four or more terms. This method involves pairing terms, factoring out common factors, and identifying shared binomials. These skills are crucial for solving equations and simplifying algebraic expressions.

Greatest Common Factor and Factoring

Greatest common factor in expressions

Top images from around the web for Greatest common factor in expressions

• 

  8.4 Greatest Common Factor and Factor by Grouping – Introductory Algebra View original

  Is this image relevant?

• 

  Greatest Common Factor and Factor by Grouping – Intermediate Algebra View original

  Is this image relevant?

• 

  8.4 Greatest Common Factor and Factor by Grouping – Introductory Algebra View original

  Is this image relevant?

• 

  8.4 Greatest Common Factor and Factor by Grouping – Introductory Algebra View original

  Is this image relevant?

• 

  Greatest Common Factor and Factor by Grouping – Intermediate Algebra View original

  Is this image relevant?

1 of 3

Top images from around the web for Greatest common factor in expressions

• 

  8.4 Greatest Common Factor and Factor by Grouping – Introductory Algebra View original

  Is this image relevant?

• 

  Greatest Common Factor and Factor by Grouping – Intermediate Algebra View original

  Is this image relevant?

• 

  8.4 Greatest Common Factor and Factor by Grouping – Introductory Algebra View original

  Is this image relevant?

• 

  8.4 Greatest Common Factor and Factor by Grouping – Introductory Algebra View original

  Is this image relevant?

• 

  Greatest Common Factor and Factor by Grouping – Intermediate Algebra View original

  Is this image relevant?

1 of 3

• Largest factor that divides all terms in an algebraic expression evenly without a remainder
• To find the GCF of an algebraic expression:
  • Factor each term completely (using prime factorization)
  • Identify common factors among all terms
  • Choose factor with highest degree (variables) or largest value (numbers) as GCF
• Examples:
  • $6x^2 + 9x$: GCF is $3x$ (common factor with highest degree)
  • $12x^3y^2 + 8x^2y$: GCF is $4x^2y$ (largest common factor)

Factoring polynomials with GCF

• Divide each term by GCF and factor it out
• Steps to factor polynomial using GCF:
  1. Identify GCF of polynomial
  2. Divide each term by GCF
  3. Write factored expression as product of GCF and quotient (result of division)
• Example:
  • $15x^3 + 25x^2$
    • GCF is $5x^2$ (common factor with highest degree)
    • Dividing each term by GCF: $15x^3 \div 5x^2 = 3x$ and $25x^2 \div 5x^2 = 5$
    • Factored expression: $5x^2(3x + 5)$ (factored form)

Factor by grouping method

• Used to factor polynomials with four or more terms
• Steps to factor by grouping:
  1. Group terms in pairs with a common factor
  2. Factor out GCF from each pair
  3. Identify common binomial factor in resulting expression
  4. Factor out common binomial
• Example:
  • $6x^3 + 9x^2 - 4x - 6$
    • Group terms: $(6x^3 + 9x^2) + (-4x - 6)$
    • Factor out GCF from each group: $3x^2(2x + 3) - 2(2x + 3)$
    • Identify common binomial factor: $(2x + 3)$
    • Factor out common binomial: $(2x + 3)(3x^2 - 2)$

Understanding algebraic expressions

• Like terms: Terms with the same variables raised to the same powers
• Coefficient: The numerical factor of a term containing a variable