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(x+y+z)^5

4 min read Jun 17, 2024
(x+y+z)^5

Expanding (x + y + z)^5: A Journey Through the Binomial Theorem

Expanding the expression (x + y + z)^5 may seem daunting at first, but it can be accomplished with the help of the Binomial Theorem. While the theorem itself deals with binomials (expressions with two terms), we can cleverly apply it to trinomials like (x + y + z) by treating it as a series of nested binomials.

The Binomial Theorem

The Binomial Theorem states that for any real numbers x and y and any non-negative integer n:

(x + y)^n = ∑(k=0)^n nCk * x^(n-k) * y^k

where nCk represents the binomial coefficient, which is calculated as:

nCk = n! / (k! * (n-k)!)

Applying the Theorem to (x + y + z)^5

  1. Grouping: We can rewrite (x + y + z)^5 as [(x + y) + z]^5. This allows us to treat (x + y) as a single term and apply the Binomial Theorem.

  2. Expansion: Using the Binomial Theorem, we expand [(x + y) + z]^5:

    (x + y + z)^5 = ∑(k=0)^5 5Ck * (x + y)^(5-k) * z^k

  3. Recursive Expansion: Now we have a series of terms where each (x + y)^(5-k) needs to be expanded using the Binomial Theorem again. For instance, (x + y)^4 would become:

    (x + y)^4 = ∑(j=0)^4 4Cj * x^(4-j) * y^j

  4. Combining Terms: After expanding all the (x + y)^(5-k) terms, we'll have a long sum with terms of the form x^a * y^b * z^c. We need to group similar terms together, adding their coefficients.

The Result

The final expansion of (x + y + z)^5 will be a sum of many terms, each containing a combination of x, y, and z raised to different powers. The specific coefficients for each term will depend on the binomial coefficients calculated during the expansion process.

While the process is tedious to perform manually, there are tools and software available to calculate the expansion efficiently. The key takeaway is that the Binomial Theorem, while designed for binomials, can be creatively applied to expand trinomials by treating them as nested binomials.

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