Question

Factor the sum or difference of two cubes. $$ 27-8 x^{3} $$

Short answer

\[(3 - 2x)(9 + 6x + 4x^2)\]

Step-by-step solution

Recognize the structure of the expression

Identify that the given expression is in the form of a difference of cubes. The general form for the difference of cubes is \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\].

Find the cube roots

Find the cube roots of both terms. \(27\) is \(3^3\) and \(8x^3\) is \((2x)^3\). Thus, \(a = 3\) and \(b = 2x\).

Apply the formula

Substitute \(a\) and \(b\) into the difference of cubes formula. \[27 - 8x^3 = 3^3 - (2x)^3 = (3 - 2x)(3^2 + 3(2x) + (2x)^2)\].

Simplify the expression

Simplify the expression inside the parentheses: \[3^2 + 3(2x) + (2x)^2 = 9 + 6x + 4x^2\].

Write the final factored form

Combine the simplified parts to get the final expression: \[27 - 8x^3 = (3 - 2x)(9 + 6x + 4x^2)\].

Key concepts

Difference of Cubes

When factoring expressions in algebra, it's important to recognize certain patterns. One common pattern is the difference of cubes, which takes the form \(a^3 - b^3\). The factoring formula associated with the difference of cubes is: \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\].

This formula works because when the two factors \(a - b\) and \(a^2 + ab + b^2\) are multiplied, they cancel out the middle terms, leaving you with \(a^3 - b^3\).

As an example, in the expression \(27 - 8x^3\), we can identify \(a = 3\) and \(b = 2x\), since \(3^3 = 27\) and \((2x)^3 = 8x^3\). Using the difference of cubes formula, we can factor this expression as \[27 - 8x^3 = (3 - 2x)(9 + 6x + 4x^2)\].

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler components (factors) that, when multiplied together, give back the original polynomial. This is a fundamental skill in algebra that helps simplify expressions and solve equations.

Consider the polynomial \(27 - 8x^3\). We identify it's a difference of cubes, but it could be polynomials in other forms too, like difference of squares or in quadratic form. Knowing the specific pattern allows us to use the right factoring technique.

The general approach to factoring polynomials includes:

• Looking for common factors.
• Identifying patterns like differences of squares, sums/differences of cubes, and trinomials.
• Applying the appropriate factoring formulas.
• Simplifying the remaining expression.

For \(27 - 8x^3\), it's the difference of cubes, and applying the formula produces factors that can't be reduced further, resulting in \((3 - 2x)(9 + 6x + 4x^2)\).

Algebraic Identities

Algebraic identities are equations that hold true for all values of the variables involved. They help in simplifying expressions and solving algebraic equations. One common identity is the difference of cubes: \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\].

These identities are invaluable because they provide a shortcut to factoring complex polynomials. Instead of expanding and simplifying line-by-line, you use the identity directly.

In the exercise \(27 - 8x^3\), we apply the difference of cubes identity:

• Identify the cubes, \a = 3\ and \b = 2x\.
• Substitute into the identity, yielding \((3 - 2x)(9 + 6x + 4x^2)\).

Because these identities are universally true, they save time and reduce errors in algebraic calculations. Recognizing and using them efficiently is a key skill in algebra.