(x-5)^4

Jasonbradley

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Jun 17, 2024

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Exploring the Expansion of (x-5)^4

The expression (x-5)^4 represents the fourth power of the binomial (x-5). Expanding this expression involves applying the binomial theorem or using a systematic approach. Let's delve into both methods.

Binomial Theorem Approach

The binomial theorem provides a general formula for expanding binomials raised to any power:

(a + b)^n = Σ (n choose k) * a^(n-k) * b^k

Where:

• n is the power to which the binomial is raised.
• k is a non-negative integer ranging from 0 to n.
• (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!).

Applying this to (x-5)^4, we have:

• a = x
• b = -5
• n = 4

Substituting these values into the formula, we get:

(x - 5)^4 = (4 choose 0) * x^4 * (-5)^0 + (4 choose 1) * x^3 * (-5)^1 + (4 choose 2) * x^2 * (-5)^2 + (4 choose 3) * x^1 * (-5)^3 + (4 choose 4) * x^0 * (-5)^4

Calculating the binomial coefficients and simplifying, we obtain the expanded form:

**(x - 5)^4 = ** x^4 - 20x^3 + 150x^2 - 500x + 625

Systematic Expansion Method

Alternatively, we can expand the expression systematically by repeatedly multiplying the binomial by itself:

(x - 5)^4 = (x - 5) * (x - 5) * (x - 5) * (x - 5)

1. First Expansion: (x - 5) * (x - 5) = x^2 - 10x + 25
2. Second Expansion: (x^2 - 10x + 25) * (x - 5) = x^3 - 15x^2 + 75x - 125
3. Third Expansion: (x^3 - 15x^2 + 75x - 125) * (x - 5) = x^4 - 20x^3 + 150x^2 - 500x + 625

This method requires careful multiplication and simplification but leads to the same result as the binomial theorem approach.

Understanding the Expanded Form

The expanded form of (x-5)^4 reveals several key points:

• Leading Term: The leading term is x^4, with a coefficient of 1. This indicates that the polynomial has a degree of 4.
• Constant Term: The constant term is 625, which is (-5)^4.
• Alternating Signs: The coefficients alternate in sign due to the presence of (-5) in the binomial.
• Symmetry: The coefficients exhibit a degree of symmetry, with the first and last terms having the same coefficient, the second and penultimate terms having the same coefficient, and so on.

Understanding these characteristics can help in analyzing and manipulating the expanded form of (x-5)^4.