Question

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

Answer 1

(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.

Answer 2

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)&sup3.

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

Answer 3

Term? Meaning each number and/or variable separated by a + or - sign. example 2x^2+4x-3 has three terms?

Answer 4

Exactly!
For example,
-3x²
is a term,
-3 is the coefficient
x is the variable,and
&sup2 is the exponent.
The sign is part of the coefficient.

Answer 5

this is crzy thing

Answer 6

"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