%5E3-expand.jpeg "(x-3)^3 Expand")
Expanding (x-3)^3
The expression (x-3)^3 represents the cube of the binomial (x-3). To expand this expression, we can use the binomial theorem or simply multiply it out step by step.
Using the Binomial Theorem
The binomial theorem states that for any positive integer n:
(a + b)^n = a^n + na^(n-1)b + (n(n-1)/2!) a^(n-2)b^2 + ... + b^n
In our case, a = x, b = -3, and n = 3. Substituting these values into the theorem, we get:
(x - 3)^3 = x^3 + 3x^2(-3) + 3x(-3)^2 + (-3)^3
Simplifying the expression, we obtain:
(x - 3)^3 = x^3 - 9x^2 + 27x - 27
Expanding by Multiplication
We can also expand (x-3)^3 by multiplying it out step by step:
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First, square the binomial: (x - 3)^2 = (x - 3)(x - 3) = x^2 - 6x + 9
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Then, multiply the result by (x-3): (x - 3)^3 = (x^2 - 6x + 9)(x - 3) = x^3 - 6x^2 + 9x - 3x^2 + 18x - 27
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Combine like terms: (x - 3)^3 = x^3 - 9x^2 + 27x - 27
Both methods lead to the same expanded form: x^3 - 9x^2 + 27x - 27. This is the expanded form of (x - 3)^3.