898 CHAPTER 12 Sequences; Induction; the Binomial Theorem Retain Your Knowledge Problems 37–45 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 37. Solve: + = x log 5 4 2 38. Solve the system: + = − − = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 4 3 7 2 5 16 39. A mass of 500 kg is suspended from two cables, as shown in the figure. What are the tensions in the two cables? 40. For = ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ A 1 2 1 0 1 4 and = − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ B 3 1 1 0 2 2 , find ⋅ A B. 41. Find the partial fraction decomposition of + − x x x 3 2 . 2 42. If a b 4, 9, = = and = c 10.2, find the measure of angle B to the nearest tenth of a degree. 43. Solve: = − e 4 x3 7 44. Find the exact value of θ tan 2 if θ = cos 5 8 and θ > sin 0. 45. If ( )( ) ( ) ( ) ( ) ′ = − + + − − f x x x x x x 2 1 3 1 2 2 , 2 2 3 find all real numbers x for which ( ) ′ = f x 0. 308 458 500 kg 12.6 The Binomial Theorem OBJECTIVES 1 Evaluate n j ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ (p. 899) 2 Use the Binomial Theorem (p. 900) Formulas have been given for expanding x a n ( ) + for n 2 = and n 3. = The Binomial Theorem * is a formula for the expansion of x a n ( ) + for any positive integer n . If n 1, 2, 3 = , and 4, the expansion of x a n ( ) + is straightforward. ( ) ( ) ( ) ( ) + = + + = + + + = + + + + = + + + + x a x a x a x ax a x a x ax a x a x a x ax a x a x a 2 3 3 4 6 4 1 2 2 2 3 3 2 2 3 4 4 3 2 2 3 4 Two terms, beginning with x1 and ending with a1 Three terms, beginning with x2 and ending with a2 Four terms, beginning with x3 and ending with a3 Five terms, beginning with x4 and ending with a4 Notice that each expansion of x a n ( ) + begins with xn and ends with a .n From left to right, the powers of x are decreasing by 1, while the powers of a are increasing by 1. Also, the number of terms equals n 1. + Notice, too, that the degree of each monomial in the expansion equals n. For example, in the expansion of x a , 3 ( ) + each monomial x ax a x a , 3 , 3 , 3 2 2 3 ( ) is of degree 3. As a result, it is reasonable to conjecture that the expansion of x a n ( ) + would look like this: ( ) + = + + + + + − − − x a x ax a x a x a n n n n n n 1 2 2 1 where the blanks are numbers to be found. This is in fact the case. Before we can fill in the blanks, we need to introduce the symbol n j . ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ *The name binomial is derived from the fact that +x a is a binomial; that is, it contains two terms.
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