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
(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
No worries! I can help you understand the question and guide you through the process.
The question is asking for the number of terms that result when you expand and simplify the expression (a+b+c)^3.
To get started, you can use the binomial theorem or the concept of distributing to expand the expression. In this case, you have three terms: a, b, and c, and each term is raised to the power of 3.
Here's how you can start:
1. Write down the expression: (a+b+c)^3.
2. Expand the expression by distributing the powers:
(a+b+c)^3 = (a+b+c)(a+b+c)(a+b+c).
At this point, you can use the distributive property to multiply the terms and combine like terms. Here's a breakdown of the process:
1. Expand the first two terms:
(a+b+c)(a+b+c) = a(a+b+c) + b(a+b+c) + c(a+b+c).
2. Apply the distributive property to each term:
= a^2 + ab + ac + ba + b^2 + bc + ca + cb + c^2.
3. Combine like terms:
= a^2 + 2ab + 2ac + b^2 + 2bc + c^2.
Now you can see that there are six terms in the expanded expression: a^2, 2ab, 2ac, b^2, 2bc, and c^2.
Therefore, after combining like terms, there are a total of 6 terms in the expansion of (a+b+c)^3.