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**Multiplying Polynomials**

Applying Generic Rectangles

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**Creating a Generic Rectangle**

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

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**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)

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**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)

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**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)

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**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 )

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**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 )

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**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

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**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

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**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

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**Using a Generic Rectangle**

Since each parentheses has 2 terms, setup a 2 X 2 rectangle: (7x + 15)(2x + 5)

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**Using a Generic Rectangle**

Label the sides with each Polynomial (7x + 15) (2x + 5) (7x + 15) (2x + 5)

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**Using a Generic Rectangle**

Multiply each row with each column (7x + 15) (2x + 5) (7x + 15) 𝟏𝟒 𝒙 𝟐 𝟑𝟎𝒙 (2x + 5) 𝟑𝟓𝒙 𝟕𝟓

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**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

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**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

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**Label the sides (3x2 – 2x + 1) (x + 3)**

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

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**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

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**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

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Factoring Trinomials ax 2 + bx + c (a > 1) Model the following by drawing rectangles with the given areas. Label the sides. Factor the area/polynomial.

Factoring Trinomials ax 2 + bx + c (a > 1) Model the following by drawing rectangles with the given areas. Label the sides. Factor the area/polynomial.

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