Problem 1 - Algebra 1: Concepts and Skills

Question

Write the three special product factoring patterns. Give an example of each pattern.

Step-by-Step Solution

Answer

The three factoring patterns are: (1) Difference of Squares: \(a^2 - b^2 = (a + b)(a -b)\), example: \(x^2 - 9 = (x + 3)(x - 3)\), (2) Perfect Square Trinomial: \(a^2 + 2ab + b^2 = (a + b)^2\) and \(a^2 - 2ab + b^2 = (a - b)^2\), examples: \(x^2 + 6x + 9 = (x + 3)^2\) and \(x^2 - 6x + 9 = (x - 3)^2\), (3) Difference and Sum of Cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) and \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\), examples: \(x^3 + 8 = (x + 2)(x^2 - 2x + 4)\) and \(x^3 - 8 = (x - 2)(x^2 + 2x + 4)\)

Step 1: Difference of Squares

This pattern can be written as \(a^2 - b^2 = (a + b)(a -b)\). An example of this pattern: \(x^2 - 9 = (x + 3)(x - 3)\)

Step 2: Perfect Square Trinomial

This pattern comes in two forms: \(a^2 + 2ab + b^2 = (a + b)^2\) and \(a^2 - 2ab + b^2 = (a - b)^2\). Examples for each form respectively: \(x^2 + 6x + 9 = (x + 3)^2\) and \(x^2 - 6x + 9 = (x - 3)^2\)

Step 3: Difference and Sum of Cubes

These patterns also come in two forms: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) and \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Examples for each form respectively: \(x^3 + 8 = (x + 2)(x^2 - 2x + 4)\) and \(x^3 - 8 = (x - 2)(x^2 + 2x + 4)\)