%5E3-expand-formula.jpeg "(x-2)^3 Expand Formula")
Expanding the Formula (x-2)³
The expression (x-2)³ represents the cube of the binomial (x-2). To expand this, we can use the binomial theorem or simply multiply the expression out.
Using the Binomial Theorem
The binomial theorem states:
(a + b)ⁿ = ∑(n choose k) * a^(n-k) * b^k
Where:
- n is the power of the binomial
- k is the index of the term (starts from 0)
- (n choose k) represents the binomial coefficient, calculated as n!/(k!(n-k)!)
Applying this to our expression:
(x - 2)³ = ∑(3 choose k) * x^(3-k) * (-2)^k
Expanding the summation:
(3 choose 0) * x³ * (-2)⁰ + (3 choose 1) * x² * (-2)¹ + (3 choose 2) * x¹ * (-2)² + (3 choose 3) * x⁰ * (-2)³
Calculating the binomial coefficients:
1 * x³ * 1 + 3 * x² * (-2) + 3 * x * 4 + 1 * 1 * (-8)
Finally, simplifying the expression:
(x - 2)³ = x³ - 6x² + 12x - 8
Expanding by Multiplication
We can also expand the expression by multiplying it out:
(x - 2)³ = (x - 2)(x - 2)(x - 2)
First, multiply the first two factors:
(x - 2)(x - 2) = x² - 4x + 4
Then, multiply the result by the remaining factor:
(x² - 4x + 4)(x - 2) = x³ - 4x² + 4x - 2x² + 8x - 8
Finally, combine like terms:
(x - 2)³ = x³ - 6x² + 12x - 8
Conclusion
Both methods result in the same expanded expression: x³ - 6x² + 12x - 8. You can choose whichever method you find easier or more convenient to use.