SECTION 12.6 The Binomial Theorem 903 Concepts and Vocabulary 12.6 Assess Your Understanding Proof ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) − ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ + ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ = − − − + − = − − + + − = − − + + − + − + − = − + + − + − + = + − + − + = + − + − + = + − + = + + − = ⎛ + ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ n j n j n j n j n j n j n j n j n j n j jn j j n j n j n j n j n j jn j n j n j n j n j jn n j n j n j n j n j j n j n n j n j n j n j n j 1 ! 1 ! 1 ! ! ! ! ! 1 ! 1 ! ! ! ! ! 1 ! 1 ! 1 ! ! 1 ! ! ! 1 ! 1 ! ! 1 ! ! 1 ! ! 1 ! ! 1 ! 1 ! ! 1 ! 1 ! 1 ! ! 1 ! 1 Multiply the first term by j j and the second term by − + − + n j n j 1 1 to obtain a common denominator. ■ Historical Feature The case = n 2 of the Binomial Theorem, ( ) +a b , 2 was known to Euclid in 300 BC , but the general law seems to have been discovered by the Persian mathematician and astronomer Omar Khayyám (1048—1131), who is also well known as the author of the Rubáiyát , a collection of four-line poems making observations on the human condition. Omar Khayyám did not state the Binomial Theorem explicitly, but he claimed to have a method for extracting third, fourth, fifth roots, and so on. A little study shows that one must know the Binomial Theorem to create such a method. The heart of the Binomial Theorem is the formula for the numerical coefficients, and, as we saw, they can be written in a symmetric triangular form. The Pascal triangle appears first in the books of Yang Hui (about 1270) and Chu Shih-chieh (1303). Pascal’s name is attached to the triangle because of the many applications he made of it, especially to counting and probability. In establishing these results, he was one of the earliest users of mathematical induction. Many people worked on the proof of the Binomial Theorem, which was finally completed for all n (including complex numbers) by Niels Abel (1802—1829). Omar Khayyám (1048–1131) 1. The is a triangular display of the binomial coefficients. 2. ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = n 0 and ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = n 1 . 3. True or False ( ) ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ = − n j j n j n ! ! ! 4. The can be used to expand expressions like ( ) +x2 3 . 6 Skill Building In Problems 5–16, evaluate each expression. 5. ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 5 3 6. ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 7 3 7. ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 7 5 8. ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 9 7 9. ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 50 49 10. ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 100 98 11. ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 1000 1000 12. ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 1000 0 13. ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 55 23 14. ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 60 20 15. ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 47 25 16. ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 37 19 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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