Polynomials

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

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)