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

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

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

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4 Find the product (3x + 1)(x + 4) using algebra tiles. Investigate You can also use a table to multiply two binomials. The model and table below show the product (3x + 1)(x + 4). x 4 3x3x 1 3x 23x 2 12x x4 Terms of the first binomial Terms of the second binomial Multiply terms to fill in the table. For example, 1 x = x. From the table, you can see that (3x + 1)(x + 4) is 3x x + x + 4, or 3x x + 4. Multiplying Binomials 12.4 LESSON

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5 Write any subtractions in the binomial as additions. Multiplying Binomials Using a Table EXAMPLE 1 Find the product (–2x + 5)(3x – 1). The product is –6x 2 + 2x + 15x – 5, or –6x x – 5. (–2x + 5)(3x – 1) = (–2x + 5)[3x + (–1)] 3x3x–1 –2x 5 –6x 2 2x2x 15x–5 First binomial Second binomial Multiplying Binomials 12.4 LESSON

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

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7 Using the Distributive Property EXAMPLE 2 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. (7x + 4)(5x + 4) = Distribute 5x + 4. = 35x x + 20x + 16 = 35x x + 16 ANSWER The area is (35x x + 16) square inches. Distribute 7x and 4. Combine like terms. SOLUTION The total length of the enlargement and mat is (7x + 4) inches. The total width is (5x + 4) inches. To find the area, multiply. 7x(5x + 4) + 4(5x + 4) Multiplying Binomials 12.4 LESSON

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8 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) = 12x 2 First + 2x Outer + 30x Inner + 5 Last Multiplying Binomials 12.4 LESSON

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

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