%5E4-expansion.jpeg "(a+b+c)^4 Expansion")
(a+b+c)^4 Expansion
The expansion of (a+b+c)^4 is a binomial expression that involves raising the sum of three terms to the power of 4. This can be achieved using the binomial theorem, which provides a formula for expanding powers of a binomial expression.
** Binomial Theorem **
The binomial theorem states that:
(a+b)^n = ∑(k=0 to n) (nCk) * a^(n-k) * b^k
where n is a positive integer, nCk is the number of combinations of n items taken k at a time, and a and b are the terms in the binomial expression.
** Expanding (a+b+c)^4 **
To expand (a+b+c)^4, we can use the binomial theorem by treating a+b as the first term and c as the second term. Then, we can apply the binomial theorem to expand the expression.
(a+b+c)^4 = ((a+b)+c)^4
= ∑(k=0 to 4) (4Ck) * (a+b)^(4-k) * c^k
Now, we can expand the expression further by applying the binomial theorem again to the (a+b) term.
(a+b+c)^4 = ∑(k=0 to 4) (4Ck) * ( ∑(j=0 to 4-k) (4-kCj) * a^(4-k-j) * b^(j) ) * c^k
** Simplifying the Expression **
After simplifying the expression, we get:
(a+b+c)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4 + 4a^3c + 12a^2bc + 12ab^2c + 4b^3c + 6a^2c^2 + 12abc^2 + 6b^2c^2 + 4ac^3 + 4bc^3 + c^4
This is the expanded form of (a+b+c)^4.
** Conclusion **
In conclusion, the expansion of (a+b+c)^4 can be achieved using the binomial theorem. The resulting expression is a polynomial with 15 terms, each with a different combination of a, b, and c. This expansion is useful in various mathematical and scientific applications, such as algebra, calculus, and physics.
