3-expand.jpeg "(2x+1)3 Expand")
Expanding (2x + 1)³
This article will guide you through expanding the expression (2x + 1)³.
Understanding the Expression
The expression (2x + 1)³ represents the product of (2x + 1) multiplied by itself three times:
(2x + 1)³ = (2x + 1)(2x + 1)(2x + 1)
Using the FOIL Method
One way to expand this expression is to use the FOIL method. FOIL stands for First, Outer, Inner, Last.
- First: Multiply the first terms of each binomial: (2x)(2x) = 4x²
- Outer: Multiply the outer terms of each binomial: (2x)(1) = 2x
- Inner: Multiply the inner terms of each binomial: (1)(2x) = 2x
- Last: Multiply the last terms of each binomial: (1)(1) = 1
Now, we have: (4x² + 2x + 2x + 1)(2x + 1)
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Combine like terms: (4x² + 4x + 1)(2x + 1)
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Repeat the FOIL method:
- First: (4x²)(2x) = 8x³
- Outer: (4x²)(1) = 4x²
- Inner: (4x)(2x) = 8x²
- Last: (4x)(1) = 4x
- First: (1)(2x) = 2x
- Outer: (1)(1) = 1
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Combine like terms again: 8x³ + 12x² + 6x + 1
Using the Binomial Theorem
Another way to expand this expression is by using the Binomial Theorem. The Binomial Theorem states that for any real numbers a and b, and any positive integer n:
(a + b)ⁿ = ∑(k=0 to n) (nCk) * a^(n-k) * b^k
Where (nCk) represents the binomial coefficient, calculated as n! / (k! * (n-k)!).
Applying the Binomial Theorem to (2x + 1)³, we get:
(2x + 1)³ = (3C0)(2x)³(1)⁰ + (3C1)(2x)²(1)¹ + (3C2)(2x)¹(1)² + (3C3)(2x)⁰(1)³
Simplifying, we get:
(2x + 1)³ = 8x³ + 12x² + 6x + 1
Conclusion
Both methods lead to the same expanded expression: 8x³ + 12x² + 6x + 1. Choose the method that is most comfortable and efficient for you.