Problem 24 - Technical Mathematics with Calculus

Question

Sum or Difference of Two Cubes. $$64 a^{3}+27$$

Step-by-Step Solution

Answer

\(64 a^{3}+27 = (4a + 3)(16a^2 - 12a + 9)\)

Step 1: Identifying the Sum of Two Cubes

Recognize the given expression as the sum of two cubes. The sum of two cubes can be factored using the formula: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here, \(64a^3 = (4a)^3\) and \(27 = 3^3\), which means \(a = 4a\) and \(b = 3\).

Step 2: Applying the Sum of Two Cubes Formula

Apply the sum of two cubes formula to factor the expression. Substituting \(a = 4a\) and \(b = 3\) into the formula \((a + b)(a^2 - ab + b^2)\) gives us \((4a + 3)((4a)^2 - (4a)(3) + 3^2)\).

Step 3: Simplifying the Factored Form

Simplify each part of the factored form. The first part is already simplified to \(4a + 3\). Simplify the second part as follows: \((4a)^2 = 16a^2\), \(-(4a)(3) = -12a\), and \(3^2 = 9\). This simplifies to \((16a^2 - 12a + 9)\). The complete factored form is \((4a + 3)(16a^2 - 12a + 9)\).

Key Concepts

Algebra

Algebra is a branch of mathematics that involves the study of symbols and the rules for manipulating these symbols to describe the number of things, relationships between quantities, and the operations performed on them. It serves as a building block for more advanced mathematics and applications in numerous fields.

One primary aspect is the expression of relationships through equations and the use of algebraic operations to solve for unknown variables. In the context of our exercise, algebra is used to identify and manipulate a sum of cubes polynomial, to factor effectively and reach a simplified expression.

Polynomial Factorization

Polynomial factorization is a process in algebra where you express a polynomial as the product of its factors, which are usually of lower degree. The factors are polynomials themselves, which when multiplied together, result in the original polynomial. This simplifies complex expressions and is especially helpful in solving equations.

For example, in our exercise, the polynomial is factored into two binomials. Recognizing the given expression as a sum of cubes is essential for the factorization process and leads to a more simplified form which is easier to evaluate or use in further calculations.

Sum of Cubes Formula

The sum of cubes formula is a special algebraic identity used to factor expressions that are the sum of two cube terms. The formula states that: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).

By using this formula, you can break down an expression like \(64a^3 + 27\) into more manageable pieces for further analysis or simplification. Factoring using the sum of cubes is an efficient technique to solve polynomials that seem daunting at first glance.

Simplifying Expressions

Simplifying expressions in algebra means to make them as compact or as straightforward as possible without changing their value. This involves combining like terms, applying the distributive property, and factoring. Simplification makes an expression more understandable and easier to work with.

In our problem, simplifying the factored form of the sum of cubes involves squaring the term \(4a\), multiplying \(4a\) by 3, and squaring the constant 3. This process transforms the factored expression into a more simplified and solvable form so that each term's contribution is transparent, making it easier to understand and apply in further problem-solving or in specific applications.