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)
It is now expanded.
We can also complete the calculation:
3 × (5+2) = 3 × 5 + 3 × 2
= 15 + 6
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 example 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 an 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 something like this?
(x + 2y)(3x − 4y)
Read Multiplying Polynomialsto find out!
Multiply by everything inside the ()
Do it in two stages:
- Write down the multiplications
- Then do the multiplications