In Math 9, I have learned about the following:
- The concept of polynomials
- Adding and subtracting polynomials
- Multiplying and dividing polynomials
- The use of algebraic tiles with polynomials
- The application of polynomials in geometry
Click on a concept to jump to it.
Note that polynomials involve exponents and fractions, which you can read about here and here respectively.
The concept of polynomials
Polynomials refer to algebraic expressions. That is to say; expressions with variables. Polynomials are made up of terms, and can be shown as “sentences” (as in, the written expression) or using shapes, also known as algebra tiles. These are the algebra tiles, coloured tiles are positive while white tiles are negative. The large square represents 
Here is an example of a polynomial in both algebraic and algebra tile forms.
The expression must be organized from the numbers with the largest amount of variables to the one with the least.
Polynomials have 4 different names, depending on the amount of terms, which are all the operations such as addition, including the amount of multiplications done with the coefficient on a variable.
Monomials are polynomials with 1 term. ie:
Binomials are polynomials with 2 terms. ie:
Trinomials are polynomials with 3 terms. ie:
Polynomials are polynomials with more than 3 terms. ie:
Polynomials also have levels of degrees. The degree is somewhat difficult to explain. It is the number of the most variables (doesn’t have to be different variable letters, just variables) in a number. This means that 
When two numbers have the same variable and exponent, they have like terms. ie:
Numbers with like terms can be added and subtracted to each other, as you will see in the next section.
Adding and subtracting polynomials
When adding and subtracting polynomials, like stated before, you can only add and subtract numbers with like terms.
In the last example, as numbers with the 
In polynomials, subtractions are treated as adding negative numbers for the purposes of simplification. When subtracting a number from the positive equivalent of that number, those numbers cancel each other. This is what’s called a zero pair.
When adding 2 different polynomials, like so:
The numbers are added or subtracted as normal.
However, when subtracting, like so:
The polynomial with the minus symbol in front of it is inverted. That is to say, the expression now looks like this:
Which is equal to:
When adding or subtracting polynomials with fractions, the numbers are added as normal.
Multiplying and dividing polynomials
When multiplying and dividing polynomials, you will need to distribute the numbers. That is to say, you will need to multiply and divide each number individually. Like terms are not needed, instead, multiplications and divisions will follow the exponent multiplication and division laws.
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When multiplying a number by another that has a variable that the first number does not have, it is added in simplification.
As with addition and subtraction, multiplication and division of fractions are done normally, still making sure to multiply/divide numbers individually. You may use the reciprocal method when dividing, and that will be shown in this example.
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5 
4
10 
4
The use of algebraic tiles with polynomials
Using algebraic tiles is a good way to visualize polynomial expression, but only smaller expressions will work. Check back to the top of the page for a review on what different tiles represent.
In this expression, there are 8 rectangles, 3 of which are negative. These rectangles cancel out 3 of the positive rectangles.
After cancelling out the rectangles, you will get this.
However, it is not organized properly. The tiles must be in descending order from left to right, like so.
Now, the expression is
To use tiles for multiplication and division, we turn to the area model. This is an area model of
When using the model, to determine the simplification, draw lines in the spaces between the rectangles to the other side of the area between all the rectangles like so.
Now the expression is simplified to
Division is largely identical, but the area component is already known, and one lineup of rectangles is missing.
In this case, simply count the amount of squares going down to find the missing number.
Multiplication and division with different variables can also be represented. In this case, they are represented using shorter blue rectangles, and the area as grey rectangles. This one being 
The area model can also be modified to work with letter representations of numbers in multiplication.
In this case, it can simply be used as a visual representation of an expression, in this case, $latex 2x(2.2+(4x+2))
The application of polynomials in geometry
Polynomials can be applied to geometry when trying to find the area or volume of a shape. For these examples, we will look at finding the area of a rectangle and triangle, then the volume of a box.
The expression to find area is: 
So the area is
Triangles are calculated identically, as they are in fact, half a rectangle. The expression is 
$latex 4x\cdot{3x+2}=12x^22x
And that is the area of the triangle.
To calculate the volume of a box you will need to multiply the length by the width by the height using 
We can confirm that a volume is the product, as 