GCF and Factoring Out GCF

Study Notes

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 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)

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)

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)

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.

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).

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).