Answer to Question #216767 in Algebra for Ann

Question

A. Expand each power using Properties I to V.

1.) (c+d)^5

2.) (x-y)^4

3.) (3-x)^4

4.) (u-v)^8

5.) (3b+y)^5

B. Find only the first three terms in the expansion.

1.) (a+20)^14

2.) (c+y)^k

C. Compute each factorial expression.

1.) 6!

2.) 8!

3.) 3!4!

Answer 1

Solution.

$A. \newline 1. \newline \left(c + d\right)^{5}=\newline=c^{5} + 5 c^{4} d + 10 c^{3} d^{2} + 10 c^{2} d^{3} + 5 c d^{4} + d^{5}. \newline 2. \newline \left(x - y\right)^{4}=\newline=x^{4} - 4 x^{3} y + 6 x^{2} y^{2} - 4 x y^{3} + y^{4}. \newline 3. \newline \left(3 - x\right)^{4}=\newline=x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81. \newline 4. \newline \left(u - v\right)^{8}=\newline=u^{8} - 8 u^{7} v + 28 u^{6} v^{2} - 56 u^{5} v^{3} + \newline+ 70 u^{4} v^{4} - 56 u^{3} v^{5} + 28 u^{2} v^{6} - 8 u v^{7} + v^{8}. \newline 5. \newline \left(3 b + y\right)^{5}=\newline=243 b^{5} + 405 b^{4} y + 270 b^{3} y^{2} + 90 b^{2} y^{3} + 15 b y^{4} + y^{5}. \newline B. \newline (k+1) \space term \space of \space binomial \space expansion \space is: \newline t_{k+1}=n C_k*(x)^{n-k}*(y)^{k}; \newline 1) \newline In \space expansion \space (a+20)^{14} \space \newline the \space first \space three \space terms \space are: \newline t_1=a^{14}; \newline t_2=280*a^{13}; \newline t_3=36400*a^{12}. \newline 2) \newline In \space expansion \space (c+y)^k \space the \space first \space three \space terms \space are: \newline t_{1}=k C_0*c^{k}*y^{0}=k C_0*c^{k}; \newline t_{2}=k C_1*c^{k-1}*y^{1}=k C_1*c^{k-1}*y; \newline t_{3}=k C_2*c^{k-2}*y^{2}. \newline C. \newline 1)\space 6!=1*2*3*4*5*6 =720. \newline 2)\space 8!=1*2*3*4*5*6*7*8=40320. \newline 3)\space 3!4!=1*2*3*1*2*3*4=6*24=144.$