91Ó°ÊÓ

The polynomial provided in the exercise is an example of a difference of cubes. In algebra, a difference of cubes is a special type of binomial that has the form \(a^3 - b^3\). The general formula for factoring a difference of cubes is: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
To use this formula, you first identify the cube roots of both terms in the polynomial. For the given exercise, we identify that \(54m^3\) can be rewritten as \((3m)^3\) and 2000 can be rewritten as \(10^3\).
You then substitute these values into the difference of cubes formula to factor the polynomial.

Polynomial factorization is the process of breaking down a polynomial into a product of simpler polynomials that, when multiplied together, give the original polynomial. Factorizing polynomials helps simplify complex expressions and solve equations.
For our exercise, after recognizing the polynomial as a difference of cubes, we use the formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
Set \(a = 3m\) and \(b = 10\), then plug these into our formula:

• \(a - b = 3m - 10\)
• \(a^2 = (3m)^2 = 9m^2\)
• \(ab = (3m)(10) = 30m\)
• \(b^2 = 10^2 = 100\)

Bringing it all together, we get:
\((3m - 10)(9m^2 + 30m + 100)\)
This is the fully factored form of the given polynomial.

In algebra, an algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. Polynomial expressions are a specific type of algebraic expression that consists of terms which are the product of constants and variables raised to whole number powers.
The given polynomial, \(54m^3 - 2000\), is an example of an algebraic expression. It involves the variable \(m\) and constants, arranged according to the rules of algebra. When dealing with such expressions, especially in factorization problems, it is crucial to identify and apply appropriate algebraic identities, like the difference of cubes.
Understanding algebraic expressions and their properties allows us to manipulate and simplify them. This aids in problem-solving and helps in deriving solutions in mathematical exercises efficiently.