Removing Group Symbols
Sometimes, algebraic expressions are written in group symbols such as:
...in which you are asked to expand and simplify as seen below:
In these cases, the distributive law is used to remove group symbols:
A binomial expression consists of two elements, for example .
A set of two binomial expressions multiplied together is called a binomial product.
Binomial products are multiplied in FOIL order, that is:
- First elements of both expressions.
- Outer elements of both expressions.
- Inner elements of both expressions.
- Last elements of both expressions.
Hence, the products are multiplied as follows:
Sometimes we see a trinomial that consists of three elements, and when multiplied it works the same way.
Several characteristics of binomial products are shown below:
This is because:
The same can be done when:
...and the proof as follows:
(Remember: two negatives multiplied will yield a positive result)
There is also a the difference of 2 squares as seen below:
Reflection: What is the difference between and ?
Expand and simplify the following:
When factorising a binomial product, take out the highest common numbers (or pronumerals) that perfectly divide both elements without leaving a remainder. This also applies to trinomials, which will be discussed further in another section.
To factorise an expression, we use the distributive law.
(Reflection: Do you remember what HCF - Highest Common Factor - means?)
Sometimes you will see expressions that are grouped in pairs, and therefore they could be factorised in pairs:
A special case of binomial products is:
There is also an interesting way to factorise the sums and differences of 2 cubes.