Vocabulary:
Degree: The largest exponent attached to a variable in an expression.
Constant: A term without a variable.
Coefficient: The whole or large number in a term.
Leading coefficient: The coefficient at the start of the expression (most left).
Binomial: When the number of terms in an expression is 3.
Trinomial: When the number of terms in an expression is 2.
Monomial: Is equivalent to one term.
How to use algebra tiles:
You will have 3 types of algebra tiles. 1, x and x². Expressions will usually have the largest amount to least amount. So the order will be in x², x then 1/whole numbers. With x² tiles, since they are the largest, they will also be physically the largest pieces. You will follow the expressions instructions. If it tells you to place 4 x’s, then place 4 x’s. But when an negative number is involved, all you need to do is flip the tiles backwards.
Add polynomials:
When starting out learning polynomials, it is good to sort the alike terms with each other. Example: 2x² + 3x + 3x². The first thing you should do is sorting the terms with the other ones that are alike. This step is not mandatory, but will help you when starting out learning to simplify polynomials. After sorted by amount, the expression should turn into 2x² + 3x² + 3x. After sorted, add the coefficients that have the same variable and exponent to each other. It should look like 5x² + 3x. Starting with small expressions is easier to learn than trying to simplify larger ones.
Subtracting polynomials:
Just like simplifying every other polynomial expression, it’s good to sort the terms. Example: 7x + 5 + 4x² – (2x² – 2 + 3x). In subtracting, all of the terms inside the bracket that is after the subtracting symbol gets flipped to its opposite number. 7x + 5 + 4x² + (-2x² + 2 – 3x). After doing this, just add and follow what the expression is. 4x² + (-2x²). 7x + (-3x). 5 – 2. The simplified version of the expression should be 2x² + 4x + 3.
Multiplying polynomials:
2(4x + 2). Multiplying polynomial expressions is almost like multiplying normally. The only difference is the variable.
Using algebra tiles for learning large multiplying expressions is very helpful. When you use algebra tiles, model the expression in a shape in a rectangle. In the example, you would take 2 small tiles and place them on the left side. On the other side, you would take 2 long tiles and 2 smaller ones and make a line with them. To simplify the expression, you must fill the space with more tiles. But, using algebra tiles and multiplying polynomials is mainly used for large expressions.
Without algebra tiles, you need to multiply normally. When there are variables, it is different. When the number that is multiplying has no variable attached, you follow whatever the variable is. In the example above, the simplified version is 8x + 4.
Some expressions might have you multiply by 2 numbers. Example: 3x + 2(4x + 2)
First, you multiply the number on the far left of the expression by the numbers inside the brackets. After that, you do the same the the number on the right; multiply them by the numbers inside the brackets. Once you have both of these numbers, you add them. The simplified version would be 12x² + 14x + 4.
Dividing polynomials:
4x + 2
_______
2
Using algebra tiles for dividing is almost pointless. Dividing polynomials is very simple if you can divide whole numbers. You would divide these numbers normally except for the variables. Just like multiplying, the same rules apply except of adding the variables, you subtract. So in the expression above, the simplified version is 2x + 1.