Factoring formulas

Variables:

Description

We use this formula to factor expressions of the form $a^2-b^2$.

Example

Example 1:| $9a^2 - 4b^2 = (3a)^2-(2b)^2=(3a+2b)(3a-2b)$| Example 2:| $102^2-4 = 102^2-2^2=(102+2)(102-2)=104 \cdot 100=10400$| Example 3:| $102 \cdot 98 = (100+2)(100-2)=100^2-2^2 = 10000-4 = 9996$

Variables:

Example

Example 1:| $Aa^3-8 &= a^3-2^3=\\ &=(a-2)(a^2+a\cdot2+2^2)=\\ &= (a-2)(a^2+2a+4)A$ | Example 2:| $A 27x^3-1 &= (3x)^3-1^3=\\ &= (3x-1)((3x)^2+3x\cdot1 +1^2)\\ &= (3x-1)(9x^2+3x+1) A$

Example

$$\begin{aligned}(a-1)^4 =&a^4-4a^3\cdot1+6a^2\cdot1^2-4a\cdot1^3+1^4\\[1.2em] =&a^4-4a^3-4a+1\end{aligned}$$

Description

This formula shows how to factor the sum of two fifth powers.