3-in-expanded-form.jpeg "(3/2x+1)3 In Expanded Form")
Expanding (3/2x + 1)³
In this article, we'll expand the expression (3/2x + 1)³ using the binomial theorem.
Understanding the Binomial Theorem
The binomial theorem provides a formula for expanding any expression of the form (a + b)ⁿ, where 'n' is a positive integer. The theorem states:
(a + b)ⁿ = aⁿ + naⁿ⁻¹b + (n(n-1)/2!)aⁿ⁻²b² + (n(n-1)(n-2)/3!)aⁿ⁻³b³ + ... + bⁿ
Where '!' represents the factorial (e.g., 5! = 5 * 4 * 3 * 2 * 1).
Applying the Theorem
Let's apply the binomial theorem to our expression (3/2x + 1)³:
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Identify 'a' and 'b':
- a = 3/2x
- b = 1
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Identify 'n':
- n = 3
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Substitute into the formula: (3/2x + 1)³ = (3/2x)³ + 3(3/2x)²(1) + 3(3/2x)(1)² + (1)³
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Simplify: (3/2x + 1)³ = 27/8x³ + 27/4x² + 9/2x + 1
The Expanded Form
Therefore, the expanded form of (3/2x + 1)³ is 27/8x³ + 27/4x² + 9/2x + 1.