View source for Polynomials, Perfect Square, Differences in Squares and Cubes

=Introduction=
So more stuff I probably know but here because I struggle with words.
=Terms=
These are mathematical expressions with consist of a Number and a variable for example 2x. The number part is called the '''coefficient'''  which is a fancy way of saying the number you need to multiply by. The variable part can be a name and raised by a power.
=Polynomial=
A polynomial is a combination of many terms. e.g 2x²+3x+24. Each term must be joined by '''addition''' or '''subtraction''' not multiplication other binary operators. There are names for terms 
*1 term - monomial
*2 terms - binomial
*3 terms - trinomial
*More than 3 - polynomial
=Examples=
So some polynomial may not show the number or the variable part. For example
 3x² + x -5
All this means is we have abbreviated the expression.
 3x² + 1x -5x⁰
There now it looks proper. The last term which is just a number is known as the '''constant term''' as it never changes.
=Degree of a term=
The degree of a term is determined by the power of the variable part so given 2x²+3x+24
- 2x² 2nd degree term
- 3x 1st degree term
- 24 constant term
Where there are 2 variable we add them together 8x²y³ is a 5 degree term<br>
<br>
People refer to the whole polynomial by the highest degree to 2x²+3x+24. So this is a 2nd degree polynomial. Polynomials are arranged by degree.
=Simplifying Polynomials=
So like terms can be added together
 2x³+2x²+5x²+10 
Can be 
 2x³+7²x+10
=Long Division=
==Examle 1==
So getting more tricky below we need to follow a process. The question is 
 (x² + 5x + 6) / (x + 2)
So we put the denominator on the left like normal long division
 
         -----------------------------
 (x + 2) | x² + 5x + 6
 
Step 1: Next we divide (x² + 5x + 6) by the first term only
           x
         -----------------------------
 (x + 2) | x² + 5x + 6

Step 2: Next we need to multiply everything in the denominator by the result (top row) x and reverse the sign
           x
         -----------------------------
 (x + 2) | x² + 5x + 6
         -(x² - 2x)

Step 3: Now calculate the remainder by take one from the other
           x
         -----------------------------
 (x + 2) | x² + 5x + 6
         -(x² - 2x)
         -----------------------------
           0  + 3x + 6

No next we do steps 1-3 again, 

Step 1: So x / 3x = + 3
           x + 3 
         -----------------------------
 (x + 2) | x² + 5x + 6
         -(x² - 2x)
         -----------------------------
           3x + 6

Step 2: Next we need to multiply everything in the denominator by the result (top row) x and reverse the sign
           x + 3 
         -----------------------------
 (x + 2) | x² + 5x + 6
         -(x² - 2x)
         -----------------------------
           3x + 6
          -(3x +6)
           -------
                0
 
Step 3: Now calculate the remainder by take one from the other give 0
 
So we can say 
 
  (x² + 5x + 6) / (x + 2) = x+3
==Example 2==
So here is the next and shows you what to do when you have something left over. Here is the question and workings

 2x³+8x²-6x+10 / x-2

    (2x²+12x+18)+ 46
    -----------------------------
x-2 | 2x³+8x²-6x+10
    -(2x³-4x²)
    ---------------
          12x²-6x+10
        -(12x²-24x)
        -----------
               18x+10
              -(18x-36)
              ---------
                     46 
So the answer is (2x²+12x+18) +  46 / x-2 left over. So back in my school days we times the lhs by (x-2)/(x-2 so we have a common denominator giving
 ((2x²+12x+18)(x-2) + 46) / x-2
 
taking the 2 out gives
 
2(x²+6x+9)(x-2) + 46 /x-2
 (x²+6x+9)(x-2) = x³-2x² + 6x²-12x + 9x-18
                = x³ + 4x² -3x -18
 
putting the 2 back
 2(x³ + 4x² -3x -18) = 2x³ + 8x² -6x -36
 
So this ends up as 
(2x³ + 8x² -6x + 10)/ (x + 2)