Factoring Patterns

Below is a reference page containing various useful factoring patterns for real numbers.

Fundamental Factoring Formulas

The formulas in this section are worth memorizing.

Perfect Square Trinomial

Perfect Square Trinomial Factoring Pattern

$$a^2+2ab+b^2=(a+b{)}^2$$

Problems

Factor each of the following expressions.

• $x^2+2x+1$
• $4y^2+12y+9$
• $a^2+10a+25$

Solution

• $x^2+2x+1=(x+1{)}^2$
• $4y^2+12y+9=(2y+3{)}^2$
• $a^2+10a+25=(a+5{)}^2$

Difference of Two Squares

Difference of Two Squares Factoring Pattern

$$a^2-b^2=(a-b)(a+b)$$

Please note that in the real numbers the sum of two squares $a^2+b^2$ is not factorable.

Problems

Factor each of the following expressions.

• $x^2-9$
• $4y^2-25$
• $a^2-49$
• $x^2+25$

Solution

• $x^2-9=(x-3)(x+3)$
• $4y^2-25=(2y-5)(2y+5)$
• $a^2-49=(a-7)(a+7)$
• $x^2+25$ is not factorable

Sum/Difference of Two Cubes

Sum/Difference of Two Cubes Factoring Pattern

$$a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)$$

The portion $a^2\mp ab+b^2$ cannot be factored further in the real numbers. The above formula is a shorthand that indicates two separate factoring patterns. For clarity, the two factoring patterns that are captured by this formula are written.

$$\{\begin{array}{l}{a}^3+b^3=(a+b)(a^2-ab+b^2)\\ a^3-b^3=(a-b)(a^2+ab+b^2)\end{array}$$

Problems

Factor each of the following expressions.

• $x^3+8$
• $27y^3-1$
• $a^3+27$

Solution

• $x^3+8=(x+2)(x^2-2x+4)$
• $27y^3-1=(3y-1)(9y^2+3y+1)$
• $a^3+27=(a+3)(a^2-3a+9)$

Factoring Beyond These Formulas

Factoring of more complicated expressions can be performed with the aid of polynomial division and complex numbers, even when the goal is to factor strictly in the real numbers.

More information on this will be added soon.