Posted by Diana on Tuesday, May 10, 2011 at 8:12am.
(a+b+c)^3
=(a+b+c)(a+b+c)(a+b+c)
= (a+b+c)(a^2 + b^2 + c^2 + 2ab + 2bc + 2ac) , I multiplied and simplified the last two factors.
= .....
See if you finish the expansion, add up all like terms and then count the number of terms.
Take (a+b)².
After expansion, it becomes:
a²+ab+ba+b²
There are physically 4 terms. But if we combine like terms,
ab+ba=2ab
So we end up with three terms:
a²+2ab+b².
The question would like an answer for the case of (a+b+c)³.
The expression is homogeneous, meaning that the sum of the exponents of every term is three.
So now we can list the ways a, b and c can be multiplied together with a sum of exponent of three, and then count the number of different terms possible.
Here's the list:
a³
a²b
a²c
ab²
abc
ac²
b³
b²c
bc²
c³
Term? Meaning each number and/or variable separated by a + or - sign. example 2x^2+4x-3 has three terms?
Exactly!
For example,
-3x²
is a term,
-3 is the coefficient
x is the variable,and
² is the exponent.
The sign is part of the coefficient.
this is crzy thing
"Like terms" are any terms in the multiplied-out product that have the same powers of a, b or c.
(a + b + c)^2 = a^2 + ab + ac + ab + b^2 + bc + ac + bc + c^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc
(a + b + c)^3 = (a + b + c)( a^2 + b^2 + c^2 + 2ab + 2ac + 2bc)
= (a^3 + ab^2 + ac^2 + 2a2b + 2 abc +2a^2c) + (a^2b + b^3 + bc^2 + 2ab^2 + 2abc + 2 b^2c) + (a^2c + b^2c + c^3 + 2abc + 2ac^2 + 2bc^2)
= a^3 + b^3 + c^3 + 6abc +3ab^2 + 3ac^2 + 3bc^2 +3a^2b + 3b^2c +3a^2c