Posted by **Diana** on Tuesday, May 10, 2011 at 8:12am.

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**, Tuesday, May 10, 2011 at 8:48am
(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**, Tuesday, May 10, 2011 at 8:48am
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**, Tuesday, May 10, 2011 at 8:52am
Term? Meaning each number and/or variable separated by a + or - sign. example 2x^2+4x-3 has three terms?

- Math -
**MathMate**, Tuesday, May 10, 2011 at 9:07am
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**, Tuesday, May 10, 2011 at 9:13am
this is crzy thing

- Math -
**drwls**, Tuesday, May 10, 2011 at 9:25am
"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

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