need help with these questions !!!! show work use numbers symbolds or words

Question

When a and b are numbers, we can expand (a+b)² as powers of a and b as follows

FoIL                                   

(a+b)2=                 

(a+b)(a+b)=

a(times)a+a(times) •b+b(times)a+b(times)b=                           

a² +2ab+b²                                              

 

1 what is the expansion for (a+b)³?    

(a+b)3 = [(a+b)(a+b)](a+b)
=[a2+2ab +b2] (a+b)
=

 

2.WHat is the corresponding expansion for (a+b)⁴ hint use (a+b)⁴ =(a+b)³ (a+b) or (a+b)⁴  =(a+b)² (a+b)²  to expand 

 

 

 

 

 

 

3. What pattern or pattern do you see that would have made this work easier ?

 

 

 

4. use the pattern(s) to find (a+b)⁵      

Answer 1

(a+b)^0=1

(a+b)^1=a+b

(a+b)^2=a^2+2ab+b^2

1. (a+b)^3=a^3+3a^2b+3ab^2+b^3

 

2. (a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4

 

3. this is the Binomial Theorem and the coefficients form Pascal's Triangle

the exponents of "a" start with 4 and go to 3, then 2, then 1, and finally 0

the exponents of "b" start with 0 and go to 1, then 2, then 3, and finally 4

the coefficient (of the next term) is found by multiplying the exponent (of the first variable, a) by its coefficient and then divided by the number of the term

look at 4a^3b... 3*4/2=12/2=6

 

 

4. (a+b)^5

    a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5

the exponents of "a" go from 5 to 0 while the exponents of "b" go from 0 to 5 (b^0=1 and a^0=1)

 

 look at the coefficients...

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

make them look like a triangle...

                                      

                                                1

                                              1  1

                                            1  2  1

                                         1  3   3   1

                                        1  4  6  4   1

                                      1  5 10 10  5  1

I don't know if it will stay a triangle but here goes...