Copyright 2013, 2010, 2007, 2005, Pearson, Education, Inc.

5.4 Special Products

The FOIL Method When multiplying 2 binomials, the distributive property can be easily remembered as the FOIL method. F – product of First terms O – product of Outside terms I – product of Inside terms L – product of Last terms

Example Multiply (y – 12)(y + 4). (y – 12)(y + 4) (y – 12)(y + 4) Product of First terms is y2 (y – 12)(y + 4) Product of Outside terms is 4y (y – 12)(y + 4) Product of Inside terms is –12y (y – 12)(y + 4) Product of Last terms is – 48 F O I L (y – 12)(y + 4) = y2 + 4y – 12y – 48 = y2 – 8y – 48

Example Multiply (2x – 4)(7x + 5). 2x(7x) + 2x(5) – 4(7x) – 4(5) F 2x(7x) F + 2x(5) O – 4(7x) I – 4(5) L (2x – 4)(7x + 5) = O I = 14x2 + 10x – 28x – 20 = 14x2 – 18x – 20 We multiplied these same two binomials together in the previous section, using a different technique, but arrived at the same product.

Special Products In the process of using the FOIL method on products of certain types of binomials, we see specific patterns that lead to special products.

Special Products Squaring a Binomial (a + b)2 = a2 + 2ab + b2 A binomial squared is equal to the square of the first term plus or minus twice the product of both terms plus the square of the second term. (a + b)2 = a2 + 2ab + b2 (a – b)2 = a2 – 2ab + b2

Example Multiply. (x + 6)2 = (x + 6)(x + 6) (x + 6)2 F O I L = x2 + 6x + 6x + 36 The inner and outer products are the same. = x2 + 12x + 36

Example Multiply. a. (12a – 3)2 = (12a)2 – 2(12a)(3) + (3)2 b. (x + y)2 = x2 + 2xy + y2

Special Products Multiplying the Sum and Difference of Two Terms The product of the sum and difference of two terms is the square of the first term minus the square of the second term. (a + b)(a – b) = a2 – b2

Example Multiply. (2x + 4)(2x – 4) = (2x)(2x) + (2x)(– 4) + (4)(2x) + (4)(– 4) F O I L = 4x2 + (– 8x) + 8x + (–16) The inner and outer products cancel. = 4x2 – 16

Example Multiply. a. (5a + 3)(5a – 3) = (5a)2 – 32 = 25a2 – 9 b. (8c + 2d)(8c – 2d) = (8c)2 – (2d)2 = 64c2 – 4d2

Example Use a special product to multiply, if possible. a. (7a + 4)2 b. (c + 0.2d)(c – 0.2d) = c2 – 0.04d2 c.