Posted by Diana on .
How many terms are in the expansion of (a+b+c)^3 after like terms have been combined?
1. I don't know what this question is asking
2 and I don't know how to start it

Math 
Reiny,
(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. 
Math 
MathMate,
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 
Math 
Diana,
Term? Meaning each number and/or variable separated by a + or  sign. example 2x^2+4x3 has three terms?

Math 
MathMate,
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. 
Math 
rosa,
this is crzy thing

Math 
drwls,
"Like terms" are any terms in the multipliedout 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