(x+y+z)^4

Jasonbradley

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Jun 17, 2024

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Expanding (x+y+z)^4

The expression (x+y+z)^4 represents the fourth power of the trinomial (x+y+z). Expanding this expression can be done using the binomial theorem or by repeated multiplication.

Using the Binomial Theorem

The binomial theorem provides a general formula for expanding expressions of the form (x + y)^n. While it's directly applicable to binomials, we can adapt it for trinomials by treating (x+y) as a single term:

1. Treat (x+y) as a single term: Consider (x+y+z)^4 as [(x+y) + z]^4.
2. Apply the binomial theorem: The binomial theorem states: (a + b)^n = Σ (n choose k) * a^(n-k) * b^k where (n choose k) is the binomial coefficient, calculated as n!/(k!(n-k)!).
3. Substitute: Substitute (x+y) for 'a' and z for 'b' in the binomial theorem formula.

This results in:

(x+y+z)^4 = [(x+y) + z]^4 =

• (4 choose 0) * (x+y)^4 * z^0 +
• (4 choose 1) * (x+y)^3 * z^1 +
• (4 choose 2) * (x+y)^2 * z^2 +
• (4 choose 3) * (x+y)^1 * z^3 +
• (4 choose 4) * (x+y)^0 * z^4

4. Expand (x+y)^n: Now, each term involves expanding (x+y) raised to a power. Again, apply the binomial theorem or use repeated multiplication for this step.

This approach systematically expands the expression and allows us to identify all the terms with their respective coefficients.

Repeated Multiplication

A more straightforward but potentially tedious method is to repeatedly multiply (x+y+z) by itself four times:

(x+y+z)^4 = (x+y+z) * (x+y+z) * (x+y+z) * (x+y+z)

Expand the expression by multiplying out each set of parentheses, combining like terms, and simplifying. This process can be lengthy but will ultimately result in the same expanded form as obtained through the binomial theorem.

Key Points

• Symmetry: The expanded form of (x+y+z)^4 will be symmetric, meaning the terms involving x, y, and z will appear in all possible combinations.
• Coefficients: The coefficients in the expanded form will be binomial coefficients, reflecting the combinations of choosing terms from the original expression.
• Number of terms: The expanded form will have 15 distinct terms, considering all possible combinations of x, y, and z.

Understanding how to expand (x+y+z)^4 is crucial in various mathematical contexts, especially when dealing with algebraic manipulations, polynomial equations, and calculus problems.