To factor binomials, the approach depends on the type of binomial you have. Here are the common methods:

1. Factoring by Taking Out the Greatest Common Factor (GCF)

• Identify the greatest common factor of both terms.
• Factor out the GCF.

Example:
x2+4xx^2+4xx2+4x
GCF is xxx, so factor it out:
x(x+4)x(x+4)x(x+4)

2. Difference of Squares

• Applies when the binomial is a difference (subtraction) of two perfect squares.
• Use the formula:

a2−b2=(a+b)(a−b)a^2-b^2=(a+b)(a-b)a2−b2=(a+b)(a−b)

Example:
x2−16x^2-16x2−16
Rewrite as x2−42x^2-4^2x2−42, so

(x+4)(x−4)(x+4)(x-4)(x+4)(x−4)

3. Sum or Difference of Cubes

• For binomials like a3+b3a^3+b^3a3+b3 or a3−b3a^3-b^3a3−b3.
• Use formulas:

a3+b3=(a+b)(a2−ab+b2)a^3+b^3=(a+b)(a^2-ab+b^2)a3+b3=(a+b)(a2−ab+b2)

a3−b3=(a−b)(a2+ab+b2)a^3-b^3=(a-b)(a^2+ab+b^2)a3−b3=(a−b)(a2+ab+b2)

Example:
x3−8=(x−2)(x2+2x+4)x^3-8=(x-2)(x^2+2x+4)x3−8=(x−2)(x2+2x+4)

4. Factoring Binomials with a Common Factor in Polynomials

• Sometimes binomials appear as common factors in polynomials.
• Factor by grouping terms and extracting the binomial as a common factor.

Example:
8x3+56x2−3x−218x^3+56x^2-3x-218x3+56x2−3x−21
Group terms: (8x3+56x2)+(−3x−21)(8x^3+56x^2)+(-3x-21)(8x3+56x2)+(−3x−21)
Factor each group: 8x2(x+7)−3(x+7)8x^2(x+7)-3(x+7)8x2(x+7)−3(x+7)
Factor out common binomial: (x+7)(8x2−3)(x+7)(8x^2-3)(x+7)(8x2−3)

Summary of Key Formulas for Special Binomials

Type| Formula| Example
---|---|---
Difference of squares| a2−b2=(a+b)(a−b)a^2-b^2=(a+b)(a-b)a2−b2=(a+b)(a−b)| x2−16=(x+4)(x−4)x^2-16=(x+4)(x-4)x2−16=(x+4)(x−4)
Sum of cubes| a3+b3=(a+b)(a2−ab+b2)a^3+b^3=(a+b)(a^2-ab+b^2)a3+b3=(a+b)(a2−ab+b2)| x3+8=(x+2)(x2−2x+4)x^3+8=(x+2)(x^2-2x+4)x3+8=(x+2)(x2−2x+4)
Difference of cubes| a3−b3=(a−b)(a2+ab+b2)a^3-b^3=(a-b)(a^2+ab+b^2)a3−b3=(a−b)(a2+ab+b2)| x3−8=(x−2)(x2+2x+4)x^3-8=(x-2)(x^2+2x+4)x3−8=(x−2)(x2+2x+4)

Steps to Factor a Binomial

1. Check if there is a greatest common factor (GCF) and factor it out.
2. Identify if the binomial fits special patterns like difference of squares or cubes.
3. Apply the appropriate formula.
4. Factor further if possible.

This approach covers most binomial factoring cases efficiently