Adding polynomials by combining like terms together and using algebra tiles is the goal of this lesson. We will show you the following three ways to add polynomials.

- Using algebra tiles
- Adding vertically
- Adding horizontally

We will start with algebra tiles since the process is a lot more straightforward and concrete with tiles. To this end, study the model below with great care.

**Example #1:**

Add 2x^{2} + 3x + 4 and 3x^{2} + x + 1**Step 1**

Model both polynomials with tiles.

**Step 2**

Combine all tiles that are alike and count them.

- You got a total of 5 light blue square tiles, so 5x
^{2}

- You got a total of 4 green rectangle tiles, so 4x

- You got a total of 5 blue small square tiles, so 5

Putting it all together, we get 5x^{2} + 4x + 5

I hope from the above modeling, it is clear that we can only combine tiles of the **same type**. For example, you could not add light blue square tiles to green rectangle tiles just like it would not make sense to add 5 potatoes to 5 apples. Try adding 5 potatoes to 5 apples and tell me if you got 10 apples or 10 potatoes. It just does not make sense!

**Example #2:**

Add 2x^{2} + -3x + -4 and -3x^{2} + -x + 1**Step 1**

Model both polynomials with tiles

**Step 2**

Combine all tiles that are alike and count them. Tiles that are alike, but have different colors will cancel each other out. We show each cancellation with a green line.

- You are left with of 1 red square tile, so -x
^{2}

- You got a total of 4 light red rectangle tiles, so -4x

- You are left with 3 strong pink small square tiles, so -3

Putting it all together, we get -x^{2} + -4x + -3

- We call tiles that are alike or are the same type "like terms"

- Like terms are terms with the same variable and the same exponent. For example, 2x
^{2}and 3x^{2}in**example #1**are like terms because they have the same variable, x and the same exponent, 2.

- To add like term, just add the coefficients, or the numbers attached to the term, or the number on the left side of the term. 2x
^{2}+ 3x^{2}= (2 + 3)x^{2}= 5x^{2}

**Example #3:**

Add 6x^{2} + 8x + 9 and 2x^{2} + -13x + 2

Line up like terms. Then add the coefficients.

6x^{2} + 8x + 9

+ 2x^{2} + -13x + 2

------------------------------

8x^{2} + -5x + 11

Example #4:

Add -5x^{3} + 4x^{2} + 6x + -8 and 3x^{3} + -2x^{2} + 4x + 12

Line up like terms. Then add the coefficients.

-5x^{3} + 4x^{2} + 6x + -8

+ 3x^{3} + -2x^{2} + 4x + 12

--------------------------------------

-2x^{3} + 2x^{2} + 10x + 4

**Example #5:**

Add 6x^{2} + 2x + 4 and 10x^{2} + 5x + 6

Combine or group all like terms. You could use parentheses to keep things organized.

(6x^{2} + 2x + 4) + (10x^{2} + 5x + 6) = (6x^{2} + 10x^{2}) + (2x + 5x) + (4 + 6)

(6x^{2} + 2x + 4) + (10x^{2} + 5x + 6) = (6 + 10)x^{2} + (2 + 5)x + (4 + 6)

Add the coefficient

(6x^{2} + 2x + 4) + (10x^{2} + 5x + 6) = 16x^{2} + 7x + 10

Example #6:

Add 2x^{4} + 5x^{3} + -x^{2} + 9x + -6 and 10x^{4} + -5x^{3} + 3x^{2} + 6x + 7

(2x^{4} + 5x^{3} + -x^{2} + 9x + -6) + (10x^{4} + -5x^{3} + 3x^{2} + 6x + 7)

= (2x^{4} + 10x^{4}) + (5x^{3} + -5x^{3}) + (-x^{2} + 3x^{2}) + (9x + 6x) + (-6 + 7)

= (2 + 10)x^{4} + (5 + -5)x^{3} + (-1 + 3)x^{2} + (9 + 6)x + (-6 + 7)

= (12)x^{4} + (0)x^{3} + (2)x^{2} + (15)x + (1)

= 12x^{4} + 2x^{2} + 15x + 1

Notice that if the term is x^{2}, you can rewrite it as 1x^{2}, so your coefficient is 1. We did this in example #6, third line with (-x^{2} + 3x^{2})

(-x^{2} + 3x^{2}) = (-1x^{2} + 3x^{2})