Rekenen met variabelen *: 3.1 Haakjes wegwerken *
Expanding brackets
Two terms and three terms
A two-term is defined as a sum (or difference) of two terms (products of constants and variables). Just as a three-term is a sum of three terms, and a polynomial a sum of an unspecified (but finite) number of terms.
The formula for the product of two two terms is the following:
As the banana-shaped loops already indicate, the right hand side is created by using a distributive property twice: \[ \begin{array}{rcl}(a+b)(c+d) &=& a(c+d) + b(c+d)\\ &=& ac +ad + bc + bd\end {array}\] Sometimes it can be useful to use a multiplication table: \[ \begin{array}{c|c|c} \cdot & c & d\\ \hline a & ac & ad\\ \hline b & bc & bd \end {array}\] After completing the table you add up the four products.
The product of two three terms
The multiplication table \[ \begin{array}{c|c|c|c} \cdot & d & e & f\\ \hline a & ad & ae & af\\ \hline b & bd & be & bf \\ \hline{c} & cd & ce & cf \end {array}\] shows that the product of two three terms \((a+b+c)(d+e+f)\) can be worked out by multiplying each term of the left-hand pair of brackets with each term in the right-hand pair of brackets and then adding up all the results: \[ \begin{array}{rcccl}(a+b+c)(d+e+f)& = &\phantom{+} ad &+ ae &+ af \\ &&+ bd &+ be &+ bf \\ &&+ cd &+ ce &+ cf\end {array}\]
We can remove the brackets using this formula. The formula can be used in all kinds of complicated situations.
The square of a binomial can be see as the product of two identical binomials:
\[\begin{array}{rclcl}\left(r+s\right)^2&=&\left(r+s\right)\left(r+s\right) &\phantom{x}&\color{blue}{\text{from square to product}}\\&=& r^2+{r}{s}+{s}{r}+s^2&\phantom{x}&\color{blue}{\text{expanding brackets}}\\&=& r^2+2r s+s^2&\phantom{x}&\color{blue}{\text{combining terms}}\end{array}\]
Or visit omptest.org if jou are taking an OMPT exam.
