1. Multiplying Polynomials
Applying Generic Rectangles

2. Creating a Generic Rectangle
WARNING: Students must have a basic understanding of Algebra Tiles to complete this tutorial.

3. Let’s start by using algebra tiles to multiply: (2x + 1)(x + 3)
The Concept
Let’s start by using algebra tiles to multiply:
(2x + 1)(x + 3)

4. Let’s start by using algebra tiles to multiply: (2x + 1) (x + 3)
The Concept
Let’s start by using algebra tiles to multiply:
(2x + 1) (x + 3)

5. Let’s start by using algebra tiles to multiply: (2x + 1) (x + 3)
The Concept
Let’s start by using algebra tiles to multiply:
(2x + 1) (x + 3)

6. Let’s start by using algebra tiles to multiply: (2x + 1) (x + 3)
The Concept
Let’s start by using algebra tiles to multiply:
(2x + 1) (x + 3)
(2x )
(x )

7. Draw in lines at every intersection to complete a rectangle:
The Concept
Draw in lines at every intersection to complete a rectangle:
(2x + 1) (x + 3)
(2x )
(x )

8. Remove the original tiles, and the answer remains:
The Concept
Remove the original tiles, and the answer remains:
(2x )
𝑥 2
𝑥 2 x
(x )
=𝟐𝐱 𝟐 +𝟕𝐱+𝟑 x x 1 x x 1 x x 1

9. The Concept (2x + 1) (x + 3) =𝟐𝐱 𝟐 +𝟕𝐱+𝟑 (2x + 1) (x + 3) =𝟐𝐱 𝟐 +𝟕𝐱+𝟑
𝑥 2
𝑥 2 x
(x )
=𝟐𝐱 𝟐 +𝟕𝐱+𝟑 x x 1 x x 1 x x 1

10. Using a Generic Rectangle
Let’s do a similar example:
(7x + 15)(2x + 5)
For this problem using tiles would be tedious and time consuming

11. Using a Generic Rectangle
Since each parentheses has 2 terms, setup a 2 X 2 rectangle:
(7x + 15)(2x + 5)

12. Using a Generic Rectangle
Label the sides with each Polynomial
(7x + 15) (2x + 5)
(7x + 15)
(2x + 5)

13. Using a Generic Rectangle
Multiply each row with each column
(7x + 15) (2x + 5)
(7x + 15)
𝟏𝟒 𝒙 𝟐
𝟑𝟎𝒙
(2x + 5)
𝟑𝟓𝒙 𝟕𝟓

14. Using a Generic Rectangle
Combine like terms and write answer
(7x + 15) (2x + 5) = 14x2 + 65x + 75
(7x + 15)
𝟏𝟒 𝒙 𝟐
𝟑𝟎𝒙
(2x + 5)
𝟑𝟓𝒙 𝟕𝟓

15. Final Example Let’s Multiply: (3x2 – 2x + 1)(x + 3)
Since there are 3 terms and 2 terms in the Parentheses, make a 3X2 rectangle

16. Label the sides (3x2 – 2x + 1) (x + 3)
Final Example
Label the sides
(3x2 – 2x + 1) (x + 3)
(3x2 – 2x + 1)
(x + 3)

17. Multiply each row with each column (3x2 – 2x + 1) (x + 3)
Final Example
Multiply each row with each column
(3x2 – 2x + 1) (x + 3)
(3x2 – 2x + 1)
3 𝒙 𝟑
-2 𝒙 𝟐 𝒙
(x + 3)
9 𝒙 𝟐
-6𝒙 𝟑

18. Combine like terms and write answer
Final Example
Combine like terms and write answer
(3x2 – 2x + 1) (x + 3) = 3x3 + 7x2 - 5x + 3
(3x2 – 2x + 1)
3 𝒙 𝟑
-2 𝒙 𝟐 𝒙
(x + 3)
9 𝒙 𝟐
-6𝒙 𝟑