43 r 2 12 r 35 24 16 x 3 44 x 2 44 x 40 25 12 n 3 20 n 2 38 n 20 26 8 b 3 4 b 2

43 r 2 12 r 35 24 16 x 3 44 x 2 44 x 40 25 12 n 3 20

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¡ 43 r 2 + 12 r ¡ 35 24) 16 x 3 + 44 x 2 + 44 x + 40 25) 12 n 3 ¡ 20 n 2 + 38 n ¡ 20 26) 8 b 3 ¡ 4 b 2 ¡ 4 b ¡ 12 27) 36 x 3 ¡ 24 x 2 y + 3 xy 2 + 12 y 3 28) 21 m 3 + 4 m 2 n ¡ 8 n 3 29) 48 n 4 ¡ 16 n 3 + 64 n 2 ¡ 6 n + 36 30) 14 a 4 + 30 a 3 ¡ 13 a 2 ¡ 12 a + 3 31) 15 k 4 + 24 k 3 + 48 k 2 + 27 k + 18 32) 42 u 4 + 76 u 3 v + 17 u 2 v 2 ¡ 18 v 4 33) 18 x 2 ¡ 15 x ¡ 12 34) 10 x 2 ¡ 55 x + 60 35) 24 x 2 ¡ 18 x ¡ 15 36) 16 x 2 ¡ 44 x ¡ 12 37) 7 x 2 ¡ 49 x + 70 38) 40 x 2 ¡ 10 x ¡ 5 39) 96 x 2 ¡ 6 40) 36 x 2 + 108 x + 81 107
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Section 3.6: Special Products Objective: Recognize and use special product rules of a sum and a dif- ference and perfect squares to multiply polynomials. There are a few shortcuts that we can take when multiplying polynomials. If we can recognize them, the shortcuts can help us arrive at the solution much faster. These shortcuts will also be useful to us as our study of algebra continues. The ±rst shortcut is often called a sum and a di±erence . A sum and a di²er- ence is easily recognized as the numbers and variables are exactly the same, but the sign in the middle is different (one sum, one difference). To illustrate the shortcut, consider the following example, where we multiply using the distributing method. Example 1. Simplify. ( a + b )( a ¡ b ) Distribute ( a + b ) a ( a + b ) ¡ b ( a + b ) Distribute a and ¡ b a 2 + ab ¡ ab ¡ b 2 Combine like terms ab ¡ ab a 2 ¡ b 2 Our Solution The important part of this example is that the middle terms subtracted to zero. Rather than going through all this work, when we have a sum and a di²erence, we will jump right to our solution by squaring the ±rst term and squaring the last term, putting a subtraction between them. This is illustrated in the following example. Example 2. Simplify. ( x ¡ 5)( x + 5) Recognize sum and di²erence Square both x and 5; put subtraction between the squares x 2 ¡ 25 Our Solution This is much quicker than going through the work of multiplying and combining like terms. Often students ask if they can just multiply out using another method and not learn the shortcut. These shortcuts are going to be very useful when we get to factoring polynomials, or reversing the multiplication process. For this reason it is very important to be able to recognize these shortcuts. More examples are shown below. 108
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Example 3. Simplify. (3 x + 7)(3 x ¡ 7) Recognize sum and di²erence Square both 3 x and 7; put subtraction between the squares 9 x 2 ¡ 49 Our Solution Example 4. Simplify. (2 x ¡ 6 y )(2 x + 6 y ) Recognize sum and di²erence Square both 2 x and 6 y ; put subtraction between the squares 4 x 2 ¡ 36 y 2 Our Solution It is interesting to note that while we can multiply and get an answer like a 2 ¡ b 2 (with subtraction), it is impossible to multiply binomial expressions and end up with a product such as a 2 + b 2 (with addition). There is also a shortcut to multiply a perfect square , which is a binomial raised to the power two. The following example illustrates multiplying a perfect square.
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