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Multiplying Binomials: Investigate and Find Product using Algebra Tiles

In this lesson, we will investigate how to find the product of binomials (3x + 1)(x + 4) using algebra tiles. We will represent each binomial using the tiles and fill in the rectangular region formed by the binomials. We will also explore different methods such as using a table and the FOIL method to find the product.

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Multiplying Binomials: Investigate and Find Product using Algebra Tiles

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  1. Multiplying Binomials 12.4 1 LESSON Investigate Find the product (3x + 1)(x + 4) using algebra tiles. Represent each binomial using algebra tiles. Arrange the first binomial vertically and the second binomial horizontally, as shown.

  2. Multiplying Binomials 12.4 2 LESSON Investigate Find the product (3x + 1)(x + 4) using algebra tiles. The binomials define a rectangular region with length (3x + 1) units and width (x + 4) units. Fill in the region with the appropriate tiles.

  3. Multiplying Binomials 12.4 3 LESSON Investigate Find the product (3x + 1)(x + 4) using algebra tiles. The tiles covering the rectangular region represent 3x2 + 13x + 4. This expression is the product of the binomials.

  4. Multiplying Binomials 12.4 LESSON Terms of the first binomial Terms of the second binomial 4 x 3x 1 Multiply terms to fill in the table. For example, 1 • x = x. Investigate Find the product (3x + 1)(x + 4) using algebra tiles. You can also use a table to multiply two binomials. The model and table below show the product (3x + 1)(x + 4). From the table, you can see that (3x + 1)(x + 4) is 3x2 + 12x + x + 4, or 3x2 + 13x + 4. 3x2 12x x 4

  5. Multiplying Binomials 12.4 LESSON 1 EXAMPLE First binomial 3x –1 Second binomial –2x 5 Multiplying Binomials Using a Table Find the product (–2x + 5)(3x – 1). Write any subtractions in the binomial as additions. (–2x + 5)(3x – 1) = (–2x + 5)[3x + (–1)] –6x2 2x 15x –5 The product is –6x2 + 2x + 15x – 5, or –6x2 + 17x – 5.

  6. Multiplying Binomials 12.4 LESSON Distributive Property Another way to multiply binomials is to use the distributive property.

  7. Multiplying Binomials 12.4 LESSON 2 EXAMPLE Using the Distributive Property Photograph You are enlarging a photo that is 7 inches long and 5 inches wide. The length and width of the enlargement are x times the length and width of the original photo. The enlargement will have a 2 inch mat. Write a polynomial expression for the combined area of the enlargement and mat. The total length of the enlargement and mat is (7x + 4) inches. The total width is (5x + 4) inches. To find the area, multiply. SOLUTION (7x + 4)(5x + 4) = 7x(5x + 4) + 4(5x + 4) Distribute 5x + 4. = 35x2 + 28x + 20x + 16 Distribute 7x and 4. = 35x2 + 48x + 16 Combine like terms. ANSWER The area is (35x2 + 48x + 16) square inches.

  8. Multiplying Binomials 12.4 LESSON First Outer Inner Last 12x2 + 2x + 30x + 5 FOIL Method Notice that using the distributive property to multiply binomials produces four products that are then added. These four products are the products of the first terms, the outer terms, the inner terms, and the last terms of the binomials. You can use the shorthand FOIL to remind you of the words First, Outer, Inner, and Last. (2x + 5)(6x + 1) =

  9. Multiplying Binomials 12.4 LESSON 3 EXAMPLE Using the FOIL Method Find the product (4x + 3)(5x – 2). First +Outer +Inner +Last (4x)(5x) + (4x)(–2) + (3)(5x) + (3)(–2) Write products of terms. Multiply. 20x2 + (–8x) + 15x + (–6) 20x2 + 7x – 6 Combine like terms.

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