(x-4)(x-4).jpeg "(x-4)(x-4)(x-4)")
Exploring the Expansion of (x-4)(x-4)(x-4)
This article will delve into the expansion of the expression (x-4)(x-4)(x-4), which can also be written as (x-4)³. We will explore the different methods to expand this expression and understand the resulting polynomial.
Understanding the Basics
Before we begin, it is important to understand that (x-4)³ represents the product of (x-4) multiplied by itself three times. This means we can expand the expression in a stepwise manner.
Method 1: Step-by-Step Multiplication
-
Expand the first two factors: (x-4)(x-4) = x² - 4x - 4x + 16 = x² - 8x + 16
-
Multiply the result by the remaining factor (x-4): (x² - 8x + 16)(x-4) = x³ - 8x² + 16x - 4x² + 32x - 64
-
Combine like terms: x³ - 8x² + 16x - 4x² + 32x - 64 = x³ - 12x² + 48x - 64
Therefore, the expanded form of (x-4)(x-4)(x-4) is x³ - 12x² + 48x - 64.
Method 2: Using the Binomial Theorem
The binomial theorem provides a formula for expanding expressions of the form (a + b)ⁿ. In our case, a = x, b = -4, and n = 3.
The binomial theorem states:
(a + b)ⁿ = ∑(n choose k) * a^(n-k) * b^k
where (n choose k) = n! / (k! * (n-k)!)
Applying this to our expression:
(x - 4)³ = (1 choose 0) * x³ * (-4)⁰ + (3 choose 1) * x² * (-4)¹ + (3 choose 2) * x¹ * (-4)² + (3 choose 3) * x⁰ * (-4)³
Simplifying:
(x - 4)³ = 1 * x³ * 1 + 3 * x² * (-4) + 3 * x * 16 + 1 * 1 * (-64)
(x - 4)³ = x³ - 12x² + 48x - 64
Conclusion
As we can see, both methods lead to the same result. Expanding (x-4)(x-4)(x-4) results in the polynomial x³ - 12x² + 48x - 64. This expression represents a cubic function with a specific shape and properties. Understanding how to expand such expressions is essential for various mathematical applications, including solving equations, analyzing graphs, and performing calculations in different fields of study.