Binomial Theorem Practice

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Formulas:

• u_{r+1}=\begin{pmatrix}n \\ r\end{pmatrix}a^{n-r}b^r
• \begin{pmatrix}n \\ r\end{pmatrix}=\frac{n!}{r!(n-r)!}

1. What is a binomial?
   Solution
   A mathematics express that consists of two terms
   
   
2. Expand without using the binomial theorem:
   1. (x-1)^2
   2. (x-1)^3
   3. (x-1)^4
   4. (2a+1)^4
      
3. Expand (x-1)^4 using the binomial theorem
   
4. Expand (x-1)^7 using the binomial theorem
   
5. Expand (2x-1)^5 using the binomial theorem
   
6. Expand (x-3y)^4 using the binomial theorem
   
7. How many terms when (x+1)^{100} is expanded out?
   Solution
   101
   
   
8. Evaluate \begin{pmatrix}7 \\ 5\end{pmatrix}
   1. With a calculator
   2. Without a calculator
      
9. Simplify:
   1. \begin{pmatrix}n \\ 0\end{pmatrix}
   2. \begin{pmatrix}n \\ 1\end{pmatrix}
   3. \begin{pmatrix}n \\ 2\end{pmatrix}
      
10. Simplify:
    1. 1^{2n-1}
    2. (-1)^{3x+1}
       
11. Use Pascal’s triangle to find the coefficients of the terms of (x+1)^6
    
12. The 5th row of Pascal’s triangle is 1, 5, 10, 10, 5, 1. Write down the values of the 6th row of Pascal’s triangle
    
13. Why is it not advisiable to use Pascal’s triangle to determine the coefficients of (2x-y)^5?
    
14. Find the coefficient of a^3b^3 in the expansion of (a+b)^5
    
15. Find the y^4 term in the expansion of (3-y)^7
    
16. In the expansion of ax^3(1+ax)^7, the coefficient of the term in x^4 is 63. Find a
    
17. Find the x^5 term in the expansion of 3x(x+1)^7
    
18. \left(x^2+\frac{3}{x}\right)^7. Find the coefficient of x^8
    
19. In the expansion of (2x-1)^n, the coefficient of the term in x^3 is 16n, where n\in\mathbb{Z}^+. Find n
    
20. The fifth term in the expansion of (x+k)^6 is 3840x^2. Find the possible values of k
    
21. \left(\frac{x^3}{2}+\frac{p}{x}\right)^8. The constant term is 7. Find the possible values of p
    
22. x^2\left(2x^2+\frac{k}{x}\right)^8. The constant term is 7168. Find the possible values of p
    
23. Given \left(1+\frac{1}{3}x\right)^n(3+nx)^2=9+45x+..., find n
    
24. \left(3x^3+\frac{b}{x}\right)^8=6561x^{24}+17496x^{20}+...+kx^0+...
    1. Find b
    2. Find k
       
25. \left(1-\frac{3}{2}x\right)^n(2+nx)^2=4-12x-45x^2+216x^3+.... Find n