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Simplifying (x + 8)³
This article explores how to simplify the expression (x + 8)³.
Understanding the Problem
The expression (x + 8)³ represents the product of (x + 8) multiplied by itself three times:
(x + 8)³ = (x + 8) * (x + 8) * (x + 8)
Simplifying Using the Binomial Theorem
One way to simplify this expression is by using the Binomial Theorem. The Binomial Theorem provides a formula to expand expressions of the form (a + b)ⁿ:
(a + b)ⁿ = aⁿ + n * aⁿ⁻¹ * b + (n * (n - 1) / 2!) * aⁿ⁻² * b² + ... + bⁿ
Where "n" is a positive integer and "!" denotes the factorial.
Let's apply this to our expression (x + 8)³:
- a = x
- b = 8
- n = 3
Substituting these values into the Binomial Theorem:
(x + 8)³ = x³ + 3 * x² * 8 + (3 * 2 / 2!) * x¹ * 8² + 8³
Simplifying further:
(x + 8)³ = x³ + 24x² + 96x + 512
Simplifying by Expanding
Another approach is to expand the expression by multiplying it out:
-
First, multiply (x + 8) by itself: (x + 8) * (x + 8) = x² + 16x + 64
-
Now, multiply the result by (x + 8): (x² + 16x + 64) * (x + 8) = x³ + 24x² + 96x + 512
As you can see, both methods lead to the same simplified expression: x³ + 24x² + 96x + 512
Conclusion
Simplifying expressions like (x + 8)³ can be done using different methods like the Binomial Theorem or expanding the expression directly. The final result is the same, providing you with a simplified polynomial.