%5E3.jpeg "(1+x)^3")
Understanding (1 + x)³
The expression (1 + x)³ represents the cube of the binomial (1 + x). In simpler terms, it means multiplying (1 + x) by itself three times:
(1 + x)³ = (1 + x) * (1 + x) * (1 + x)
Expanding the Expression
To fully understand the expression, we need to expand it. This can be done using the distributive property of multiplication or by using the binomial theorem:
Using the distributive property:
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Multiply the first two factors: (1 + x) * (1 + x) = 1 * (1 + x) + x * (1 + x) = 1 + x + x + x² = 1 + 2x + x²
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Multiply the result by the third factor: (1 + 2x + x²) * (1 + x) = 1 * (1 + x) + 2x * (1 + x) + x² * (1 + x) = 1 + x + 2x + 2x² + x² + x³
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Combine like terms: 1 + x + 2x + 2x² + x² + x³ = 1 + 3x + 3x² + x³
Using the binomial theorem:
The binomial theorem provides a general formula for expanding expressions of the form (a + b)ⁿ:
(a + b)ⁿ = ∑(k=0 to n) (n choose k) * a^(n-k) * b^k
Where (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!).
Applying this to (1 + x)³, we get:
(1 + x)³ = (3 choose 0) * 1³ * x⁰ + (3 choose 1) * 1² * x¹ + (3 choose 2) * 1¹ * x² + (3 choose 3) * 1⁰ * x³
Calculating the binomial coefficients and simplifying:
(1 + x)³ = 1 + 3x + 3x² + x³
The Result
Therefore, the expanded form of (1 + x)³ is 1 + 3x + 3x² + x³. This expression is a polynomial of degree 3, meaning it has a highest power of x equal to 3.
Applications
The expression (1 + x)³ has applications in various areas of mathematics, including:
- Calculus: It's used in finding derivatives and integrals of functions.
- Algebra: It helps in understanding the expansion of binomial expressions.
- Probability and Statistics: It appears in formulas for calculating probabilities and moments.
Understanding the expansion of (1 + x)³ is essential for comprehending more complex mathematical concepts and problem-solving in various fields.