Question 146046
I will do the first few modules for you here. Try doing some on your own via these examples and post the remaining modules if you still require help.

Module 1:
10. <=, >=, <, > 
This is commonly used notation to make statements about the relative size or order of two quantities or objects. 
"a<=b" means that a is less than or equal to b.
"a>=b" means that a is greater than or equal to b.
"a _<_ b" means that a is strictly less than b. (Take out the _--- issue with this site)
"a>b" means that a is strictly greater than b.

Further example:
(2n)!>(n-1)! for n>=1 can be proven easily by mathematical induction.


19. function
"The mathematical concept of a function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output"). A function associates a single output to each input element drawn from a fixed set, such as the real numbers." (Wikipedia)

Example:
f(x)=e^(x)sin(x)

Module 2.
5. difference of squares
These terms define themselves: a difference of squares is a difference of squared terms. Consider:
a^2-b^2
This is factored easily as a^2-b^2=(a-b)(a+b) because the non-square term is nonexistent (it is canceled).

15. factor by grouping
This phrase refers to a process of grouping terms in order to factor generally higher degreed polynomials than quadratics. Simply, determine if two pairs of terms have something the same in common. If they do, they can be factored out.
Example:
x^3-3x^2+x-3=x^3+x-3x^2-3=x(x^2+1)-3(x^2+1)=(x-3)(x^2+1)