%5E3-expanded-form.jpeg "(x-1)^3 Expanded Form")
Expanding (x - 1)^3
The expression (x - 1)^3 represents the product of (x - 1) multiplied by itself three times: (x - 1)(x - 1)(x - 1). To find its expanded form, we can use the following methods:
1. Direct Multiplication:
We can expand the expression step by step using the distributive property:
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First step: Multiply the first two factors: (x - 1)(x - 1) = x(x - 1) - 1(x - 1) = x^2 - x - x + 1 = x^2 - 2x + 1
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Second step: Multiply the result by the third factor: (x^2 - 2x + 1)(x - 1) = x^2(x - 1) - 2x(x - 1) + 1(x - 1) = x^3 - x^2 - 2x^2 + 2x + x - 1 = x^3 - 3x^2 + 3x - 1
2. Using the Binomial Theorem:
The binomial theorem provides a formula for expanding expressions of the form (a + b)^n. In this case, a = x, b = -1, and n = 3.
The binomial theorem states:
(a + b)^n = a^n + (n choose 1)a^(n-1)b + (n choose 2)a^(n-2)b^2 + ... + (n choose n-1)ab^(n-1) + b^n
Where (n choose k) is the binomial coefficient, calculated as:
(n choose k) = n! / (k! (n-k)!)
Applying this to our problem:
(x - 1)^3 = x^3 + (3 choose 1)x^2(-1) + (3 choose 2)x(-1)^2 + (-1)^3
Simplifying:
(x - 1)^3 = x^3 - 3x^2 + 3x - 1
Conclusion:
The expanded form of (x - 1)^3 is x^3 - 3x^2 + 3x - 1. Both direct multiplication and the binomial theorem provide ways to achieve this result.