GCF and Factoring Out GCF

Greatest Common Factor (GCF)

• The GCF is the largest number that divides two or more integers without leaving a remainder.
• To find the GCF of a polynomial, list the factors of each term and identify the common factors.
• The GCF can be a single variable, a constant, or a combination of variables and constants.

Factoring Out Greatest Common Factor

• Factoring out the GCF involves dividing each term of the polynomial by the GCF.
• This is useful for simplifying polynomials and making them easier to work with.
• Example: Factor out the GCF from 6x^2 + 12x
  • GCF = 6x
  • Factored form: 6x(x + 2)

Factoring Quadratic Expressions

• A quadratic expression is a polynomial of degree two, in the form ax^2 + bx + c.
• Factoring quadratic expressions involves expressing them as a product of two binomials.
• There are several methods for factoring quadratic expressions, including:
  • Factoring out the GCF
  • Using the "ac method" (where a and c are the coefficients of the x^2 and constant terms)
  • Using the " guessing method" (where you try different combinations of factors)
• Example: Factor the quadratic expression x^2 + 5x + 6
  • Factored form: (x + 3)(x + 2)

Factoring Sums and Differences of Cubes

• A sum of cubes is a polynomial in the form a^3 + b^3.
• A difference of cubes is a polynomial in the form a^3 - b^3.
• Factoring sums and differences of cubes involves using the following formulas:
  • a^3 + b^3 = (a + b)(a^2 - ab + b^2)
  • a^3 - b^3 = (a - b)(a^2 + ab + b^2)
• Example: Factor the sum of cubes x^3 + 8
  • Factored form: (x + 2)(x^2 - 2x + 4)

Greatest Common Factor (GCF)

• The GCF is the largest number that divides two or more integers without leaving a remainder.
• To find the GCF of a polynomial, list the factors of each term and identify the common factors.
• The GCF can be a single variable, a constant, or a combination of variables and constants.

Factoring Out Greatest Common Factor

• Factoring out the GCF involves dividing each term of the polynomial by the GCF.
• This process simplifies polynomials and makes them easier to work with.
• Example: Factoring out the GCF from 6x^2 + 12x results in 6x(x + 2).

Factoring Quadratic Expressions

• A quadratic expression is a polynomial of degree two, in the form ax^2 + bx + c.
• Factoring quadratic expressions involves expressing them as a product of two binomials.
• Methods for factoring quadratic expressions include:
  • Factoring out the GCF.
  • Using the "ac method" (where a and c are the coefficients of the x^2 and constant terms).
  • Using the "guessing method" (where you try different combinations of factors).
• Example: Factoring the quadratic expression x^2 + 5x + 6 results in (x + 3)(x + 2).

Factoring Sums and Differences of Cubes

• A sum of cubes is a polynomial in the form a^3 + b^3.
• A difference of cubes is a polynomial in the form a^3 - b^3.
• Factoring sums and differences of cubes involves using 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: Factoring the sum of cubes x^3 + 8 results in (x + 2)(x^2 - 2x + 4).