Expanding brackets
If you expand brackets you use the distributive law.
A bracket is multiplied with a factor
When a bracket is expanded, each term within the bracket is multiplied by the factor outside the bracket. Here, + and - are set according to the sign rules. The following applies:
$\color{red}{a}\cdot(b+c)=\color{red}{a}\cdot b+\color{red}{a}\cdot c$
Examples
- $5 \cdot (x+y)=5\cdot x+5\cdot y$
- $-2\cdot (5x-2)$ $=-2\cdot 5x+(-2)\cdot (-2)$ $=-10x+4$
- $(1-y)\cdot x=x\cdot 1+x\cdot (-y)$ $=x-xy$
A bracket is multiplied with a bracket
If you expand two brackets, each member of one bracket is multiplied by each member of the other bracket. Here, + and - are set according to the sign rules. The following applies:
$(\color{red}{a}+\color{blue}{b})\cdot(c+d)=\color{red}{a}c+\color{red}{a}d+\color{blue}{b}c+\color{blue}{b}d$
Examples
- $(4a+3b)\cdot(5x−8y)$ $=20ax−32ay+15bx−24by$
-
$(2a-4b)\cdot(-7a-2b+9)$
$=-14a^2-4ab+18a+28ab$ $+8b^2-36b$
$=-14a^2+24ab+18a+8b^2-36b$