Algebra - Expanding

"Expanding" means removing the ( ) ... but we have to do it the right way!

( ) are called "parentheses" or "brackets"

Whatever is inside the ( ) needs to be treated as a "package".

So when multiplying: multiply by everything inside the "package".

 

Example: Expand 3 × (5+2)

3 × (5+2) = 3 × 5 + 3 × 2

It is now expanded.

We could also go on to calculate that it equals 15 + 6 = 21

In Algebra

In Algebra putting two things next to each other usually means to multiply.

So 3(a+b) means to multiply 3 by (a+b)

Here is an examle of expanding, using variables a, b and c instead of numbers:

And here is another example involving some numbers. Notice the "·" between the 3 and 6 to mean multiply, so 3·6 = 18:

Multiplying negatives has special rules: a negative times a positive gives a negative, but multiplying two negatives gives a positive:

In that case −3 · -5 = +15 (a positive answer), but here is another example where the second part is negative:

So the second term ended up negative because 2x · −a = −2ax, (it is also neater to write "2ax" rather than "2xa").

That was also interesting because of x being squared (x2)

Lastly, we have an example with three terms inside:

The same rule applies: multiply by everything inside the ().

And here is a hint: when a multiplication is obvious (like a · 2) do it straight away, but when it needs more thought (like a · −b) leave it for the next line.

Many Times Many

How do we do this?

(x + 2y)(3x − 4y)

Read Multiplying Polynomialsto find out!

Conclusion

Multiply by everything inside the ()

Do it in two stages:

  • Write down the multiplications
  • Then do the multiplications