%5E3-expand.jpeg "(x-3)^3 Expand")
Expanding (x-3)^3
In mathematics, expanding an expression like (x-3)^3 means to multiply it by itself three times and then simplify the result. This process involves the use of the binomial theorem and some algebraic manipulations.
The Binomial Theorem
The binomial theorem is a mathematical formula that describes the expansion of powers of a binomial, which is an expression consisting of two terms. The formula is given by:
$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
where $a$ and $b$ are the two terms, $n$ is the power to which the binomial is raised, and $\binom{n}{k}$ is the binomial coefficient, which can be calculated as:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
Expanding (x-3)^3
Using the binomial theorem, we can expand (x-3)^3 as follows:
$(x-3)^3 = \sum_{k=0}^3 \binom{3}{k} x^{3-k} (-3)^k$
Simplifying the expression, we get:
$(x-3)^3 = \binom{3}{0} x^3 (-3)^0 - \binom{3}{1} x^2 (-3)^1 + \binom{3}{2} x (-3)^2 - \binom{3}{3} (-3)^3$
Evaluating the binomial coefficients, we get:
$(x-3)^3 = x^3 - 3x^2 \cdot 3 + 3x \cdot 3^2 - 3^3$
Simplifying further, we get:
$(x-3)^3 = x^3 - 9x^2 + 27x - 27$
And that's the result!
Conclusion
Expanding (x-3)^3 involves applying the binomial theorem and simplifying the resulting expression. The final result is a cubic polynomial in x, with coefficients that can be calculated using the binomial theorem.
