## Problem Set 42: Binomial Theorem

1. What is a binomial coefficient, and how it is calculated?

2. What role do binomial coefficients play in a binomial expansion? Are they restricted to any type of number?

3. What is the Binomial Theorem and what is its use?

4. When is it an advantage to use the Binomial Theorem? Explain.

For the following exercises, evaluate the binomial coefficient.

5. $\left(\begin{array}{c}6\\ 2\end{array}\right)$

6. $\left(\begin{array}{c}5\\ 3\end{array}\right)$

7. $\left(\begin{array}{c}7\\ 4\end{array}\right)$

8. $\left(\begin{array}{c}9\\ 7\end{array}\right)$

9. $\left(\begin{array}{c}10\\ 9\end{array}\right)$

10. $\left(\begin{array}{c}25\\ 11\end{array}\right)$

11. $\left(\begin{array}{c}17\\ 6\end{array}\right)$

12. $\left(\begin{array}{c}200\\ 199\end{array}\right)$

For the following exercises, use the Binomial Theorem to expand each binomial.

13. ${\left(4a-b\right)}^{3}$

14. ${\left(5a+2\right)}^{3}$

15. ${\left(3a+2b\right)}^{3}$

16. ${\left(2x+3y\right)}^{4}$

17. ${\left(4x+2y\right)}^{5}$

18. ${\left(3x - 2y\right)}^{4}$

19. ${\left(4x - 3y\right)}^{5}$

20. ${\left(\frac{1}{x}+3y\right)}^{5}$

21. ${\left({x}^{-1}+2{y}^{-1}\right)}^{4}$

22. ${\left(\sqrt{x}-\sqrt{y}\right)}^{5}$

For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.

23. ${\left(a+b\right)}^{17}$

24. ${\left(x - 1\right)}^{18}$

25. ${\left(a - 2b\right)}^{15}$

26. ${\left(x - 2y\right)}^{8}$

27. ${\left(3a+b\right)}^{20}$

28. ${\left(2a+4b\right)}^{7}$

29. ${\left({x}^{3}-\sqrt{y}\right)}^{8}$

For the following exercises, find the indicated term of each binomial without fully expanding the binomial.

30. The fourth term of ${\left(2x - 3y\right)}^{4}$

31. The fourth term of ${\left(3x - 2y\right)}^{5}$

32. The third term of ${\left(6x - 3y\right)}^{7}$

33. The eighth term of ${\left(7+5y\right)}^{14}$

34. The seventh term of ${\left(a+b\right)}^{11}$

35. The fifth term of ${\left(x-y\right)}^{7}$

36. The tenth term of ${\left(x - 1\right)}^{12}$

37. The ninth term of ${\left(a - 3{b}^{2}\right)}^{11}$

38. The fourth term of ${\left({x}^{3}-\frac{1}{2}\right)}^{10}$

39. The eighth term of ${\left(\frac{y}{2}+\frac{2}{x}\right)}^{9}$

For the following exercises, use the Binomial Theorem to expand the binomial $f\left(x\right)={\left(x+3\right)}^{4}$. Then find and graph each indicated sum on one set of axes.

40. Find and graph ${f}_{1}\left(x\right)$, such that ${f}_{1}\left(x\right)$ is the first term of the expansion.

41. Find and graph ${f}_{2}\left(x\right)$, such that ${f}_{2}\left(x\right)$ is the sum of the first two terms of the expansion.

42. Find and graph ${f}_{3}\left(x\right)$, such that ${f}_{3}\left(x\right)$ is the sum of the first three terms of the expansion.

43. Find and graph ${f}_{4}\left(x\right)$, such that ${f}_{4}\left(x\right)$ is the sum of the first four terms of the expansion.

44. Find and graph ${f}_{5}\left(x\right)$, such that ${f}_{5}\left(x\right)$ is the sum of the first five terms of the expansion.

45. In the expansion of ${\left(5x+3y\right)}^{n}$, each term has the form $\left(\begin{array}{c}n\\ k\end{array}\right){a}^{n-k}{b}^{k}, \text{where} k$ successively takes on the value $0,1,2,…,n$. If $\left(\begin{array}{c}n\\ k\end{array}\right)=\left(\begin{array}{c}7\\ 2\end{array}\right)$, what is the corresponding term?

46. In the expansion of ${\left(a+b\right)}^{n}$, the coefficient of ${a}^{n-k}{b}^{k}$ is the same as the coefficient of which other term?

47. Find $\left(\begin{array}{c}n\\ k - 1\end{array}\right)+\left(\begin{array}{c}n\\ k\end{array}\right)$ and write the answer as a binomial coefficient in the form $\left(\begin{array}{c}n\\ k\end{array}\right)$. Prove it. Hint: Use the fact that, for any integer $p$, such that $p\ge 1,p!=p\left(p - 1\right)!\text{.}$

48. Consider the expansion of ${\left(x+b\right)}^{40}$. What is the exponent of $b$ in the $k\text{th}$ term?

49. Which expression cannot be expanded using the Binomial Theorem? Explain.

• $\left({x}^{2}-2x+1\right)$
• ${\left(\sqrt{a}+4\sqrt{a}-5\right)}^{8}$
• ${\left({x}^{3}+2{y}^{2}-z\right)}^{5}$
• ${\left(3{x}^{2}-\sqrt{2{y}^{3}}\right)}^{12}$