How do you find the binomial expansion for #(2x+3)^3#?

2 Answers
Jul 10, 2015

#(2x+3)^3 = 8x^3 + 36x^2 + 54x + 27#

Explanation:

With the Pascal's triangle, it's easy to find every binomial expansion :

Each term, of this triangle, is the result of the sum of two terms on the top-line. (example in red)

#1#
#1. 1#
#color(blue)(1. 2. 1)#
#1. color(red)3 . color(red)3. 1#
#1. 4. color(red)6. 4. 1#
...

More, each line has the information of one binomial expansion :

The 1st line, for the power #0#
The 2nd, for the power #1#
The 3rd, for the power #2#...

For example : #(a+b)^2# we will use the 3rd line in blue following this expansion :

#(a+b)^2 = color(blue)1*a^2*b^0 + color(blue)2*a^1*b^1 + color(blue)1*a^0*b^2#

Then : # (a+b)^2 = a^2 + 2ab + b^2#

To the power #3# :

#(a+b)^3 = color(green)1*a^3*b^0 + color(green)3*a^2*b^1 + color(green)3*a^1*b^2 + color(green)1*a^0*b^3#

Then #(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3#

So here we have #color(red)(a=2x)# and #color(blue)(b=3)# :

And #(2x+3)^3 = color(red)((2x))^3 + 3*color(red)((2x))^2*color(blue)3 + 3*color(red)((2x))*color(blue)3^2 + color(blue)3^3#

Therefore : #(2x+3)^3 = 8x^3 + 36x^2 + 54x + 27#

Jul 10, 2015

#(2x+3)^3=8x^3+36x^2+54x+27#

Explanation:

#(2x+3)^3#

Use the cube of a sum method, in which #(a+b)^3=a^3+3a^2b+3ab^2+b^3#.

#a=2x;# #b=3#

#(2x+3)^3=(2x)^3+(3*2x^2*3)+(3*2x*3^2)+3^3# =

#8x^3+(3*4x^2*3)+(3*2x*9)+27# =

#8x^3+(9*4x^2)+(27*2x)+27# =

#8x^3+36x^2+54x+27#