Question

Expand completely.

11) (2m − 1)^4

12) (x − y)^3

13) (x^4 − y)^5

14) (2x^3 + 1)^5

15) (y − x^2)^3

16) (y^3 − 4x)^3

Answer 1

11)

(2m − 1)^4  =  [ (2m − 1)^2 ] X [(2m − 1)^2 ]

                   =  [ 4m^2 - 4m + 1 ] X [ 4m^2 - 4m + 1 ]

                   =  (4m^2)(4m^2 - 4m + 1) - 4m(4m^2 - 4m + 1) + 1(4m^2 - 4m + 1)

                   =  (16m^4 - 16m^3 + 4m^2) - (16m^3 - 16m^2 + 4m) + (4m^2 - 4m + 1)

                   =  16m^4 - 16m^3 + 4m^2 - 16m^3 + 16m^2 - 4m + 4m^2 - 4m + 1

                   =   16m^4 - 32m^3 + 24m^2 - 8m + 1

 

12)

(x − y)^3  =  x^3 - 3(x^2)y + 3xy^2 - y^3

 

13)

(x^4 − y)^5  =  [ (x^4 − y)^2 ] X [ (x^4 − y)^3 ]  

                   =  [ (x^8 − 2(x^4)y + y^2 ] X [ x^12 -3(x^8)y + 3(x^4)y^2 - y^3 ] 

 

                   =  x^8[ x^12 -3(x^8)y + 3(x^4)y^2 - y^3 ]− 2(x^4)y[ x^12 -3(x^8)y + 3(x^4)y^2 - y^3 ]+ (y^2)[ x^12 -3(x^8)y + 3(x^4)y^2 - y^3 ] 

 

                   =  [ x^20 -3(x^16)y + 3(x^12)y^2 - (x^8)y^3 ] − [ 2(x^16)y -6(x^12)y^2 + 6(x^8)y^3 - 2(x^4)y^4 ] + [ (x^12)y^2 -3(x^8)y^3 + 3(x^4)y^4 - y^5 ] 

 

                   =  x^20 -3(x^16)y + 3(x^12)y^2 - (x^8)y^3 − 2(x^16)y + 6(x^12)y^2 - 6(x^8)y^3 + 2(x^4)y^4 + (x^12)y^2 -3(x^8)y^3 + 3(x^4)y^4 - y^5 

 

                   =  x^20 - 5(x^16)y + 10(x^12)y^2 - 10(x^8)y^3 + 5(x^4)y^4 - y^5 

 

14)

(2x^3 + 1)^5  = [ (2x^3 + 1)^2 ] X [ (2x^3 + 1)^3 ]

                      = [ 4x^6 + 4x^3 + 1 ] X [ (8x^9 + 12x^6 + 6x^3 + 1) ]

 

                      = 4x^6(8x^9 + 12x^6 + 6x^3 + 1) + 4x^3(8x^9 + 12x^6 + 6x^3 + 1) + 1(8x^9 + 12x^6 + 6x^3 + 1)

 

                      = 32x^15 + 48x^12 + 24x^9 + 4x^6 + 32x^12 + 48x^9 + 24x^6 + 4x^3 + 8x^9 + 12x^6 + 6x^3 + 1

 

                      = 32x^15 + 80x^12 + 80x^9 + 40x^6 + 10x^3 + 1

    

15)

(y − x^2)^3  =  y^3 -3(y^2)x^2 + 3yx^4 - x^6

 

16)

(y^3 − 4x)^3  =  y^9 - 3(y^6)(4x) + 3(y^3)(16x^2) - (4x)^3

                     =  y^9 - 12(y^6)x + 48(y^3)x^2 - 64x^3

 

11) (2m − 1)^4  =  16m^4 - 32m^3 + 24m^2 - 8m + 1

12) (x − y)^3  =  x^3 - 3(x^2)y + 3xy^2 - y^3

13) (x^4 − y)^5  =  x^20 - 5(x^16)y + 10(x^12)y^2 - 10(x^8)y^3 + 5(x^4)y^4 - y^5 

14) (2x^3 + 1)^5   = 32x^15 + 80x^12 + 80x^9 + 40x^6 + 10x^3 + 1

15) (y − x^2)^3  =  y^3 -3(y^2)x^2 + 3yx^4 - x^6

16) (y^3 − 4x)^3   =  y^9 - 12(y^6)x + 48(y^3)x^2 - 64x^3.