Problem 64 - Algebra A Combined Function

Question

Factor. $$ 64-p^{3} $$

Step-by-Step Solution

Answer

\((4 - p)(16 + 4p + p^2)\)

Step 1: Recognize the Expression

The given expression is \(64 - p^3\). This is a binomial expression that can be recognized as a difference of cubes, since \(64\) is a perfect cube.

Step 2: Identify the Cube Terms

We know that \(64 = 4^3\). So, the expression \(64 - p^3\) can be rewritten as \((4)^3 - (p)^3\), which is a difference of cubes.

Step 3: Apply the Difference of Cubes Formula

The formula for factoring a difference of cubes \(a^3 - b^3\) is \((a - b)(a^2 + ab + b^2)\). Here, \(a = 4\) and \(b = p\).

Step 4: Substitute and Simplify

Substitute \(a = 4\) and \(b = p\) into the formula to get \((4 - p)((4)^2 + 4p + p^2)\).

Step 5: Perform Calculations

Calculate \((4)^2 + 4p + p^2\) which simplifies to \(16 + 4p + p^2\). So the expression becomes \((4 - p)(16 + 4p + p^2)\).

Key Concepts

Difference of Cubes

The difference of cubes is a special type of algebraic expression that simplifies to a well-known pattern. In mathematics, a cube refers to raising a number or variable to the power of three. When you have an expression of the form \(a^3 - b^3\), it's considered a difference of cubes. Since both terms are cubes, they imply certain symmetry that allows for streamlined simplification using a specific formula.
Recognizing these expressions is vital because they follow a particular factorization pattern. The formula for factoring a difference of cubes is given by:

• \((a - b)(a^2 + ab + b^2)\)

This formula highlights that a difference of cubes can be split into two factors: a binomial and a trinomial. The challenge is often to rewrite the expression in terms of cubes and apply this pattern efficiently. This makes seemingly complex algebraic expressions simpler to handle and solve.

Binomial Expression

A binomial expression in algebra is an algebraic expression that contains exactly two terms. These terms are usually connected by either a plus or a minus sign. Binomials are often encountered in various algebraic problems, including factoring and polynomial expansions.
In the case of \(64 - p^3\), the expression stands as a binomial. Here, \(64\) and \(-p^3\) are the two distinct terms, connected by a minus sign. Recognizing a binomial expression is crucial because many algebraic formulas apply directly to them, enabling further simplification.
Understanding binomials and how they fit into certain algebraic patterns, such as the difference of cubes, can significantly ease the process of solving complex expressions. By identifying and recognizing these forms, students can apply appropriate formulas more quickly and accurately.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation symbols that represent mathematical constructs. Unlike simple arithmetic expressions, algebraic expressions often involve variables, which makes them versatile and applicable in a wide range of problems.
Algebraic expressions can be simplified, factored, or expanded based on the operations involved. In equations like \(64 - p^3\), you deal with both constants (like \(64\)) and variables (\(p\)) raised to powers. This specific expression falls into the category of a binomial, as it has two terms.
To work with algebraic expressions effectively, identifying patterns such as the difference of cubes, or other polynomial identities, can lead to direct simplification. This insight is crucial in both simplifying complex expressions and solving equations.

Algebraic Formulas

In algebra, formulas are pre-determined equations or identities that help solve particular problems efficiently. These formulas are derived from fundamental mathematical principles and can simplify complex calculations by providing a direct path to the solution.
For example, the difference of cubes formula is:

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

Algebraic formulas like these are essential tools in a mathematician’s toolkit. They simplify the process of breaking down complicated expressions into manageable parts, allowing for easier resolution of equations or simplifications.
Understanding and applying these formulas not only makes the process quicker but also minimizes errors that can occur with complex algebraic manipulations. Mastery of these formulas is key to success in algebra and beyond.