Common pitfalls in Math 121
1. Distributive Property
Did you know that other than multiplication nothing distributes over addition?
The Distributive property says: c(a + b) = ca + cb
That is c(a + b) = ca + cb and c(a – b) = ca – cb since subtraction is really addition of
opposites.
· Roots Do Not Distribute over addition: √a + b ≠ √a + √b
· Powers Do Not Distribute over addition: (a + b)n ≠ an + bn, n ≠ 1
·
Reciprocals Do Not
distribute over addition:
· Logs Do Not Distribute over addition: log (a + b) ≠ log a + log b
2. Main Principle of addition
This principle says that the only things you can add are SAME THINGS.
That is, a of something + b of something = (a + b) of something.
For example 3x + 4x = 7x, or 3dogs + 4 dogs = 7 dogs.
Can you add 3x + 4y? The answer is no, they are not same things.
Can you answer the following questions?
When can I add fractions?
When can I add polynomials?
When can I add radicals?
When can I add anything? When I understand what SAME THINGS means for anything. You get the picture.
3. main principle of multiplication
It says that you can multiply anything by anything.
4. Simplifying fractions
What does it mean to simplify a fraction?
Is
?
What did I do? I cancelled the x’s and the 4’s.
When it comes to simplifying fractions canceling is Bad Language. We do not cancel things to simplify fractions.
To simplify a fraction we factorize a 1.
That is, given a fraction, factor the numerator and the denominator and remove any unnecessary 1’s.
For example:
Can I simplify
?
That is, can I factorize a 1? Let us see.
So
the answer is no.