Problem 61 - College Algebra Essentials

Question

Factor using the formula for the sum or difference of two cubes $$8 x^{3}-1$$

Step-by-Step Solution

Answer

The factored form of \(8x^3 - 1\) is \((2x + 1)(4x^2 - 2x + 1)\).

Step 1: Identify a and b

Express the given expression as a difference of cubes. In this case, \(8 x^{3}\) can be expressed as \((2x)^3\) and \(-1\) can be expressed as \((-1)^3\). So, \(a = 2x\) and \(b = -1\).

Step 2: Apply the Formula for Difference of Two Cubes

Plug \(a\) and \(b\) into the formula for the difference of two cubes \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Substituting \(a = 2x\) and \(b = -1\) gives \((2x - (-1))( (2x)^2 + (2x)(-1) + (-1)^2)\).

Step 3: Simplify the Result

Simplify the expression. Using the order of operations (PEMDAS/BODMAS), the expression simplifies to \((2x + 1)(4x^2 - 2x + 1)\).

Key Concepts

Difference of Cubes

The concept of "difference of cubes" refers to an algebraic expression involving two perfect cubes separated by subtraction. It's a special form of polynomial with a straightforward formula that makes it easy to factor. When you see an expression like \(a^3 - b^3\), you can apply the formula:

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

This formula helps us break down the expression into two smaller polynomial expressions. To use it, first identify what is being cubed in your expression so you can assign values to \(a\) and \(b\). In our example, \(8x^3 - 1\), it becomes clear that \(a = 2x\) and \(b = -1\) since \((2x)^3 = 8x^3\) and \((-1)^3 = -1\). Once \(a\) and \(b\) are identified, substituting these into the formula makes the factoring process straightforward. Always ensure to fully simplify your results.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the building blocks of algebra, allowing you to represent real-world problems in mathematical terms. These expressions can consist of constants, coefficients, and variables. For example, in the expression \(8x^3 - 1\), \(8x^3\) is a term where 8 is a coefficient and \(x\) is a variable raised to the power of 3.When dealing with algebraic expressions, especially in factoring, understanding the components of each term is essential. This knowledge allows you to apply specific formulas effectively, like the difference of cubes, to simplify and solve expressions. Simplification and factoring help not only with solving problems but also with recognizing patterns and relationships within the expressions themselves.

Polynomial Equations

Polynomial equations are equations involving polynomials, where the expression is equal to a particular value, usually zero. A polynomial itself is an algebraic expression consisting of terms that are made up of variables raised to exponents and coefficients. Recognizing a polynomial equation is crucial, especially when solving or factoring them.To solve polynomial equations, some common steps include factoring the polynomial, using formulas for special cases like sum or difference of squares, and simplifying the expression to find the roots. In our given example, factoring \(8x^3 - 1\) using the difference of cubes formula resulted in the polynomial being expressed as \((2x + 1)(4x^2 - 2x + 1)\). Like this, solutions to polynomial equations often involve rewriting them in a simpler, more workable form to find where the equation holds true. Understanding these techniques is vital for skills in algebra and further mathematical applications.