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)
   -----------------------------

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.

Example 3

So the final example was

6x⁴-9x²+18 / x-3

The trick to this question to fill in the missing terms first

6x⁴+0x³-9x²+0x¹+18 / x-3

Filling in these means you can apply exactly the same steps

Greatest Common Factor (GCF)

This is where you look at a polynomial and find, you guest it, the greatest common factor so given

3x + 15

It would be 3 because

3(x + 5)

And

4x² + 8x = 4x(x+2) GCF = 4x
5x² -15x³ = 5x²(1 -3x) GCF = 5x²

Factoring by Grouping

Here we divide the polynomial into two group and take out the GCF

x³ - 4x² + 3x -12

So group one

x³ - 4x² = x²(x-4)

Group two

3x -12 = 3(x - 4)

So they both have (x-4) so we can say

(x-4)(x² + 3)

Next example

  2r³ -6r² + 5r - 15
= 2r²(r -3) +5(r -3)
= (r-3)(2r² +5)

Factoring by Trinomials

For these you look for numbers which multiply to the constant and add up to the coefficient of the x term

x² + 3x - 28

So in the case two numbers which multiply to -28 and add up to +3

So 28 factors are 1, 28, 2, 14, 7, 4 - so 7 and 4

(x -4)(x+7)

And given

x² - 3x - 10

So factors of 10 are 1,10,2,5 - so 5 and 2

(x-5)(x+2)

When the first term is not x² for example

2x² + 20x + 48

Just get rid of the 2

2(x² + 10x + 24)
2(x+4)(x+6)

What do you do I you cannot factor out to x². A tiny bit more complex this bit because the slight of hand of the youtuber. You need to multiply the coefficient of the first term with the constant and then look for two numbers which when added together = the middle term. So given

2x² - 5x + -3

So 2x -3 = -6 and factors of -6 are 2, -3, 1 -6
So 1 -6 = -5 = the middle term coefficient<br And this was the tricky bit. Just replace p-5x with -6x +1x. It is obvious when you see it but they mean replace the middle term with the factors you identified. In this case 1 and -6. So we have

2x² - 6x + 1x -3

So using the grouping approach take out the GCF we have

2x(x-3) + 1(x-3)
(x-3)(2x+1)

Perfect Square Trinomial

Intro

This was the bane of my life today and hopeful will be clear as day from now on. The perfect square trinomial is of the form

A² + 2AB + B² = (A+B)²

And this is the bit I needed. If you take square root term 1, and term 3 and multiply them by 2 and it equals term 2 you are golden.

So for example

x² + 8x + 16

Using the approach from above rather than perfect square, by grouping we can see factors of 16 which = 8 are 4,4 and so

(x+4)(x+4) = (x+4)²

Using the perfect square approach we do

Square root of x² = x
Square root of 16 = 4
Middle term is 2(4x)

Not a Perfect Square

So here is one which is not. This was the tutors step by step

9x² + 6x + 4

Square root of 9x² = 3
Square root of 4   = 2
3x2 = 6 but we have to double it because it is 2(AB) = 12 so not a perfect square

Is a Perfect Square

And a good one

4x² + 12x + 9

Square root of 4x² = 2
Square root of 9   = 3
3x2 = 6 but we have to double it because it is 2(AB) = 12 so is a perfect square
Looking what makes 4x² is 2x and what makes 9 is 3 so answer is
(2x+3)²

Difference in Squares

So if we want to do difference in square A²-B² = (A+B)(A-B). Needed to understand the (A+B)(A-B) so

 = (A+B)(A-B)
 = A² - AB + BA + B² 

Now -AB + AB = 0 So

 = A² - B² 

So looking at some examples

x² - 25 = (x+5)(x-5)
x² - 36 = (x+6)(x-6)
4x² - 9 = (2x+3)(2x-3)