I was tutoring my brother for calculus 2 the other day on the trapezoidal rule for area approximation, the following function:√(x2+1) when I noticed he was making a consistent horrible mistake...
He kept pulling out the +1 out of the square root to try to simplify this. So he claimed, √(1 +1)= √1+√1= 1+1 = 2.
An obvious problem with this is the claim that√2 = 2, when we know that 2×2=4 not 2.
The other problem with this is that it violates the distributive property. What do you mean Jehsuamo? These are
roots we are dealing with the distributive property is about addition and multiplication isn't it?
Yes the distributive property is a statement about addition and multiplication, but what gets lost with the root symbol may easily be seen with rational exponents.
If you already know that a(b+c) = ab +ac,
and are fine with multiplying binomials, you should know that (a+b)2 is not a2+b2
So...
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Let's use our imagination a bit. Picture yourself in math class (Algebra I to be exact), minding your own business, having fun playing with the axioms (aka rules) of algebra, and then one day your teacher drops this bomb on you:
"Expand (x+3)(x-1)"
And you might be thinking, "woah now, where did come from?"
It makes sense that this would shock you. You were just getting used to the idea of expanding 3(x-1), and you probably would have been fine with x+3(x-1), but (x+3)(x-1) is a foreign idea all together.
Well, before you have much time to think about it on your own and discover anything interesting, your teacher will probably tell you that even though you don't know how to solve it now, there is a "super helpful", magical technique that will help you…
FOIL
For those of you lucky enough never to have heard of FOIL, I will explain. FOIL stands for First Outside Inside Last and is a common mnemonic...
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