1. Polynomials and Polynomials Operations
Grade 9 Math
Bedford Junior High

2. Polynomials: Monomials, Binomials, and Trinomials
A monomial is a constant or numerical coefficient, a variable or literal coefficient , or the product of a constant and one or more variables.
A polynomial is the sum of one or more monomials.
"... this means that a monomial is a single term which has NO positive sign (+) or negative sign (-) between entries.  It would be things like:
 13, 3x, -57, x², 4y²,
-2xy, or  520x²y²   but NOT (2x+7) or (4x²-2x)."

3. "A polynomial can be one monomial or a number of monomials grouped together with positive signs and/or negative signs.”
The expression  3x² is a monomial because it has one term.
The expression 5x² + 2x is a binomial because it has two terms.
The expression  6x² +7x + 8 is a trinomial because it has three terms.
All three expressions can be called polynomials. 

4. Operations with Polynomials
Adding
Add like terms by adding the numerical coefficients.   
ex.    4ab + -2ab = 2ab
Subtracting
Subtract one monomial from another like monomials, add the opposite of the second number. ex. 4ab - (-2ab) =
4ab +(+2ab)=6ab        
ex.    6x2 - x2 = 5x x2 +(- x2) = 5x2

5. Operations with Polynomials
Multiply monomials: Multiply the numerical coefficients.
When variable factors are powers with the same base, multiply by adding exponent
ex.    3x2(4x3) = 12x5
Because (3)(4) = 12
and ( x2)(x3) = x (2+3) = x5
Therefore x2(4x3) = 12x5
Divide monomials:
Divide their numerical coefficients.
When variable factors are powers with the same base, divide by subtracting exponents. ex.     30x21
(6x4) = 5x17
( 30 ÷ 6) = 5 and
x21 ÷ x4 = x17
Therefore, the answer is 5x17

6. Distributive Property
Means that what is inside the bracket is affected by the outside term.
For example with a sharing model:
3 ( x + 2) this means we have 3 groups of ( x+ 2) Which would give us a total of 3x +6 v

7. Distributive Property
Area Model
Remember that Area = length X width
3( x +2) the (l) is 3 and (w) is (x+2)
Therefore 3( x +2) = 3x + 6

8. Distributive Property
Symbolically :
The “3” affects each item inside the bracket. It does not matter how many terms are inside the bracket. Each term is affected
3 ( x+ 2)
3(x) + 3(2)
3x + 6
Another example is 7x( 2x + 4y):
(7x) (2x) + (7x) (4y)
(7)(2)(x)(x) + (7)(4)(x)(y)
14 x xy

9. Factoring Polynomial Expressions Symbolically ( Finding the GCF)
Let’s say we have the following expression
35 a a a
we have to look for the greatest numerical coefficient and literal coefficient that can come out of each part of the expression.
(7·5·a·a·a) - (4·7·a·a) + (3·7·a)
Therefore, we find the factors of each term. “7” is common to all three terms as well as “a”. What is left over from the term goes inside the bracket. Our GCF is 7a . The leftovers are ( 5a a + 3) . So our two factors are: ( 7a) ( 5a2 - 4a + 3)

10. Factoring : We are now back to where we started.
We can use the distributive principle and the area model to check our answer. Our length is (7a) and our width is ( 5a2 - 4a + 3) . Therefore our area is 35 a a a
We are now back to where we started.
5a2
- 4a 3 7a
35a3
-28a2
+21a

11. Multiplying Two Binomials
When building a rectangular tile model, use the following guidelines:
Begin at the top left corner with the x2 tiles ( Region 1)
Construct a rectangle in the bottom right corner with the unit tiles ( Region 4)
Fill the top right and bottom left with the x-tiles
( regions 2 and 3)
If you draw a rectangle with a length of (x + 1) and width of
( x + 5) the area of the rectangle is :
x2 +6x +5
Region 1 x2
Region 2
X tiles
Region 3
x tiles
Region 4
Unit tiles

12. Multiplying Two Binomials Using the Distributive Principle
Using the distributive principle you can do this symbolically as follows:
(x+3)(x+5)
(x)(x+5) +3(x+5)
(x)(x) + (x)(5) + (3)(x) + (3)(5)
x2 + 5x +3x + 15 ( Remember to combine like terms)
x2 + 8x ( final simplified version)

13. Another Way to Remember the Distributive Principle
(x+3)(x+5)
Multiply the FIRST terms in each bracket
Multiply the first term in the first set of brackets and the last term in the section set of brackets (OUTSIDE)
multiply the last term in the first set of brackets and first term in the second set of brackets (INSIDE)
multiply the last terms in each set of brackets. ( LAST)
An acronym is FOIL
( x) ( x) = x2
(x) (5) = 5x
(3)(x) = 3x
( 3) (5) = 15
Therefore we get
x2 + 8x + 15

14. Multiplying Binomials with Negatives
If you have one side of the rectangle positive and the other side negative, then using the distributive principle
(x -2) ( x +3)
X2 + 3x -2x -6
You need the pieces to complete all regions of your rectangle , but your final outcome will be :
X2 + x - 6 because you use the ZERO Principle
However, to have a complete rectangle , you NEED the pieces + 2x and -2x

15. Factoring Trinomials 1,6 7 -1,-6 -7 2,3 5 -2,-3 -5 Sum of the Factors
When you are factoring trinomials , you are organizing your tiles into a rectangle . Its side measures become your factors.
One way to help you find your side measures is to look at your unit tiles
If we were factoring
X2 +7x +6
We need to find two numbers whose sum is 7 and whose product is 6.
We list the pairs of factors of 6. We add each pair of factors .
Because we know that x times x is x2,
then our first part of the term in each bracket is ( x ) (x )
We look for the factors that have the sum of 7 . They are 1 and 6.
We write these numbers as the second terms in the binomials.
X2 +7x +6 = ( x + 1 ) ( x + 6)
Factors of 6
Sum of the Factors
1,6 7
-1,-6 -7
2,3 5
-2,-3 -5

16. More Factoring -1,6 +5 1, -6 -5 -2 , 3 +1 2,-3 -1 Factors of - 6
If it was X2 +x -6 then it would have to be
( x - 2 ) ( x + 3) because -2 time 3 equals -6 and -2 plus 3 equals +1
One side of your rectangle must be positive and the other side must be negative.
Two positive and two negative create Zero , so we will have to put in two positive x’s and two negative x’s to complete the rectangle.
Factors of - 6
Sum of the Factors
-1,6 +5
1, -6 -5
-2 , 3 +1
2,-3 -1

17. Dividing Trinomials
If you are dividing a trinomial , then you can use the area model to help you find your answer.
If you have x2 + 4x + 4 and you are dividing it by (x + 2) , you make one of your side measures (x + 2).
Now you can start organizing your tiles into a rectangle .
If one side is (x+2) then the other side MUST be (x+2)
x2 + 4x + 4 = (x+2)(x+2)
( x + 2)
= ( x + 2)