Maths Times Infinity > Topics > Algebra
Algebra
Algebra is incredibly useful has some very general applications but this will cover some simple factorisation and expansion of expressions. Also how to solve equations.
Expansion
Let's say we have 3(x+2y). This can be "expanded" to 3x+6y. An example where you might actually use this logic is if you wanted to buy 23 hats and each one costed 7. First you might find the price of 20 hats by doubling 7 to get 14 and then timesing that by 10 to get 140. Then you would add on the remaining 3 hats which give 21. So your total would be 161. You could write this as #7(23) = 7(20+3) = 140+21 = 161#.
A harder kind of expansion is like this: #(x+h)(2g-t)#. You can break it up into two steps if you like. First you get #x(2g-t) + h(2g-t)# by expanding the first brackets. Then you expand these to get #2xg-xt+2hg-ht# and that's your answer. Notice that there are 4 terms now.
Factorisation
This is the reverse of expanding. Basically you have to notice a common factor or a pattern and then group the terms together. For example, can you see a common factor in 6y2 - 30x? Well, there's a factor of #2#. So we can write #2(3y^2-15x)#. Any more factors? Yep, #3#'s still a factor. Now we get #6(y^2-5x)#.
Difference of Two Squares
This comes up a lot. It's kind of a trick, a nice little factorisation.
Check it out:
\begin{align} (x+y)(x-y) &= x^2-xy+yx-y^2 \\ &= x^2-y^2 \end{align}Woah, did you see that? Read it again in slow-motion. Like how those two middle terms cancelled out? Pretty wicked-awesome, huh? Some examples:
#36-49h^2 = (6+7h)(6-7h)#.
#(g+3)(g-3) = g^2-9#.
Let's say you wanted to find 47*53:
\begin{align} 47*53 &=(50-3)(50+3) \\ &= 50^2-3^2 \\ &= 2500-9 \\ &= 2491 \end{align}