Expand (1 + 2x - 1/2)^2 and (3x + 1/3)^3?

Answer 1

Answer 1:
(1 + 2x - 1/2)^2
= 1 + (2x)^2 + (-1/2)^2 + 2(2x)(1) + 2(2x)(-1/2) + 2(-1/2)(1)
= 1 + 4x^2 + 1/4 + 4x -2x -1
= 4x^2 + 2x + 1/4

Answer 2:
(3x + 1/3)^3
= (3x)^3 + (1/3)^3 + 3(3x)(1/3)[3x + 1/3]
= 27x^3 + 1/27 +3x(3x + 1/3)
= 27x^3 + 1/27 + 9x^2 + x
= 27x^3 + 9x^2 + x + 1/27

Answer 2

(1 + 2x - 1/2) = 2x + 1/2
so (2x+1/2)^2 = 4x^2 + 2x + 1/4
[since (ax+b)^2 =a^2x^2 + 2abx + b^2..here a = 2 and b= 1/2]

(3x + 1/3 )^3
[(a+b)^3 = a^3+b^3+3ab(a+b)]
with a= 3x and b= 1/3 , ab = x
(3x + 1/3 )^3 = (3x)^3 + (1/3)^3 + 3x (3x+1/3)
=27x^3 + 1/27 + 9x^2 + x
rearranging
= 27x^3 + 9x^2+ x + 1/27

Answer 3

(1 + 2x - 1/2)(1+2x-1/2)

Simplify - 1 - 1/2 = 1/2

(2x +1/2)(2x+1/2)

Factorise - 4x^2 + x + x + 1/4

Simplify

Answer = 4x^2 + 2x + 1/4

(3x + 1/3)^3

(3x + 1/3)(3x + 1/3)(3x + 1/3)

Work out the first two -

(3x + 1/3)(3x + 1/3)

9x^2 + x + x + 1/9 = 9x^2 + 2x + 1/9

9x^2 + 2x + 1/9(3x + 1/3)

27x^3 + 6x^2 + 3/9(1/3)

Simplify -

27x^3 + 6x^2 + 1/3(1/3)

Answer = 9x^3 + 2x^2 + 1/9

Answer 4

(a+b)^2 = a^2 + 2ab + b^2
(1 + 2x - 1/2)^2 = (2x + 1/2)^2 = 4x^2 + 2x + 1/4
(a+b)^3 = (a+b)(a^2 - ab + b^2)
(3x + 1/3)^3 = (3x+1/3)(9x^2 - x + 1/9)
remember : (9x^2 - x + 1/9)>0

Answer 5

(1+2x-1/2)^2=[(1-1/2)+2x]^2
=(1/2+2x)^2
=(1/2)^2+2(1/2)(2x)+(2x)^2 {(a+b)^2=a^2+2ab+b^2}
=1/4+2x+4(x^2)

(3x+1/3)^3=(3x)^3+(1/3)^3+3(3x)(1/3)(3x+1/3)
because (a+b)^3=a^3+b^3+3ab(a+b)
=27(x^3)+1/27+3x(3x+1/3)
=27(x^3)+1/27+9(x^2)+x
=27(x^3)+9(x^2)+x+1/27

Answer 6

(1+2x-1/2)^2
=(1+4x^2+1/4+4x-2x-1)
=(4x^2+2x+1/4)
(3x+1/3)^3
=(27x^3+9x^2+3x+1/27)