Problem 58 - Elementary and Intermediate Algebra

Question

Factor difference of cubes. \(125 m^{3}-x^{6}\)

Step-by-Step Solution

Answer

The factored form of \(125m^3 - x^6\) is \((5m - x^2)(25m^2 + 5mx^2 + x^4)\).

Step 1: Identify it as a difference of cubes

A difference of cubes formula is expressed as \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). We need to recognize that \(125m^3 - x^6\) is in this form. We recognize \(125m^3\) as \((5m)^3\) and \(x^6\) as \((x^2)^3\).

Step 2: Identify terms a and b

From our recognition in Step 1, set \(a = 5m\) and \(b = x^2\).

Step 3: Write the factored formula

Substitute \(a = 5m\) and \(b = x^2\) into the formula for the difference of cubes. This gives:\( (5m)^3 - (x^2)^3 = (5m - x^2)((5m)^2 + (5m)(x^2) + (x^2)^2)\).

Step 4: Simplify the expression

Perform the arithmetic inside the parentheses:- \((5m)^2 = 25m^2\)- \((5m)(x^2) = 5mx^2\)- \((x^2)^2 = x^4\) Substitute these back into the expression to get:\((5m - x^2)(25m^2 + 5mx^2 + x^4)\).

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra. It involves breaking down a complex polynomial into simpler factors or products. This process is similar to finding factors of a number in arithmetic but applies to algebraic expressions. By factoring polynomials, we can simplify expressions, solve equations, and understand mathematical concepts more clearly.

Here are some basic steps involved in factoring polynomials:

• Identify any common factors in the polynomial.
• Determine the polynomial's special form, like a difference of squares or cubes.
• Apply the appropriate method or formula to factor the polynomial completely.

In our example, we identified the original polynomial expression as a difference of cubes, which leads us to use the specific algebraic formula suited for this type of expression.

Algebraic Identities

Algebraic identities are equations that hold true for all values of the variables involved. In factoring problems, certain identities allow us to transform complex expressions into more manageable forms.

The difference of cubes is a common algebraic identity used when the expression fits the pattern \(a^3 - b^3\). This identity is expressed as:

• \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)

By matching our polynomial to this identity, we simplify the expression into a product of binomials.


Knowing and applying these identities makes solving algebra problems more efficient by recognizing patterns and simplifying calculations.

Polynomial Expressions

A polynomial expression is a sum of terms, each consisting of a coefficient, a variable, and a non-negative integer exponent. Understanding polynomial expressions involves recognizing their specific structure and behavior.

For instance, the expression in the exercise, \(125 m^{3}-x^{6}\), consists of two terms that form a binomial. By rewriting the individual terms as cubes, it reveals that the expression could be considered a difference of cubes through the transformation:

• \(125 m^{3} = (5m)^3\)
• \(x^{6} = (x^2)^3\)

Grasping the structure of polynomials helps in employing the right technique, whether it's factoring, expanding, or simplifying expressions as needed in different mathematical scenarios.