Question

Explain what a polynomials is and identify the different parts of a polynomial.

Answer 1

Question 1 
Polynomials are expressions that include constants and variables. We can use addition, subtraction, multiplication and we can use exponents but only with positive integers. In polynomials, you cannot use division by a variable. This means that you can't have terms like:
$\frac{a}{x^n}$
Where a is constant, x is a variable and n is a positive integer.
Here is an example of a polynomial:
$5x^3+3x^2+\frac{x}{2}+9$
x would be is a variable, please keep in mind that you can have polynomials with more than one variable. Numbers before variables are called constants.
Here are some example of constants:
$2;2/3;\pi;\sqrt(2)$
And we also have exponents. Exponents are numbers written in the upper-right corner of a variable. These must be positive integers.
Question 2
We can name polynomials based on their degree and number of terms.
A degree is largest exponent. Examples:
$x^3+2x+3; degree=3\\ x^2+2; degree=2\\ 21;degree=0$
You can have polynomial without variable.
Here is a chart showing you special names for polynomials based on their degree:
$0\ constant\\ 1 \ linear\\ 2 \ quadratic\\ 3 \ cubic\\ 4 \ quartic\\ 5 \ quintic$
There are special names for polynomials with 1, 2, 3 terms. These are monomial, binomial, trinomial. For polynomials that have more than 3 terms, we simply say polynomial of n terms.
Question 3
When adding or subtracting polynomials we simply subtract/add like terms.
Like terms are those that have the same exponent. Here is an example:
$4x^3+2x^2+x+5\\ x^4 -x^3+3x^2$
Let us add these two polynomials:
$4x^3+2x^2+x+5\\ x^4 -x^3+3x^2\\ x^4+(4x^3-x^3)+(2x^2+3x^2)+x+5=x^4+3x^3+5x^2+x+5$
Question 4
Foil stands for First, Outer, Inner, Last.
Foil can be used only to multiply two binomials( polynomials that have 2 terms). 
Here is an example:
$(2x+3)(x-2)= (2x\cdot x) (First)+(2x\cdot (-2)) (Outer)+(3\cdot x)(Inner)\\ + (3\cdot (-2)) (Last);\\ (2x+3)(x-2)=2x^2-2x+3x-6=2x^2+x-6$
Question 5
First special case is square of a sum/difference. This happens when we want to multiply two identical polynomials. Example:
$(x+2)(x+2)\\ (x^2-x)(x^2-x)$
Second special case is a product of a sum and a difference. 
Here is an example:
$(x-2)(x+2)\\ (x^2-x)(x^2+x)$
Question 6
For these two special cases we can use following formulas:
$(a\pm b)^2=a^2 \pm2ab+b^2\\ (a+b)(a-b)=a^2-b^2$
Let me explain first formula:
$(x+3)^2=x^2+6x+9\ Using the formula\\ (x+3)(x+3)=x^2+3x+3x+9=x^2+6x+9;Using FOIL$
$(x-3)^2=x^2-6x+9\ Using the formula\\ (x-3)(x-3)=x^2-3x-3x+9=x^2-6x+9;Using FOIL$
Here is an example for second formula:
$(x-3)(x+3)=x^2-9=x^2-(3)^2; Using formula\\ (x-3)(x+3)=x^2+3x-3x-9=x^2-9=x^2-(3)^2$
You can see that we get the same result, so when you have these special cases you can use formulas as a schortcut.