Question

rewrite the following expression in expanded form and simplify completely.
\\(x + y)^6\\

Explanation:

Apply the Binomial Theorem formula

$$ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k $$

For \(n = 6\):

$$ (x + y)^6 = \sum_{k=0}^{6} \binom{6}{k} x^{6-k} y^k $$

Determine the binomial coefficients using Pascal's Triangle

The coefficients for \(n = 6\) are:

$$ \binom{6}{0}=1, \quad \binom{6}{1}=6, \quad \binom{6}{2}=15, \quad \binom{6}{3}=20, \quad \binom{6}{4}=15, \quad \binom{6}{5}=6, \quad \binom{6}{6}=1 $$

Write the fully expanded and simplified expression

$$ (x + y)^6 = 1x^6y^0 + 6x^5y^1 + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6x^1y^5 + 1x^0y^6 $$

$$ (x + y)^6 = x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6 $$

Answer:

\(x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6\)