Problem 41 - College Algebra

Question

For the following exercises, factor the polynomials. $$ 64 x^{3}-125 $$

Step-by-Step Solution

Answer

The polynomial factors as \((4x-5)(16x^2 + 20x + 25)\).

Step 1: Recognize the Form

The polynomial given is a difference of cubes, which is of the form \(a^3 - b^3\). In this case, \(64x^3\) is the cube of \((4x)\) and \(125\) is the cube of \((5)\). Thus, the expression \(64x^3 - 125\) fits into this format with \(a = 4x\) and \(b = 5\).

Step 2: Apply the Difference of Cubes Formula

The difference of cubes formula is \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Substitute \(a = 4x\) and \(b = 5\) into the formula: \((4x - 5)((4x)^2 + (4x)(5) + 5^2)\).

Step 3: Simplify the Expressions Inside the Parentheses

Calculate each component. First, \((4x)^2 = 16x^2\). Then compute \((4x)(5) = 20x\), and finally, \(5^2 = 25\).

Step 4: Assemble the Factored Form

Substitute the simplified expressions back into the factored form: \((4x-5)(16x^2 + 20x + 25)\). This is the factored form of the original polynomial.

Key Concepts

Difference of Cubes

In algebra, the concept of the **difference of cubes** is a useful pattern for factoring certain types of polynomials. It specifically allows us to factor expressions of the form \(a^3 - b^3\). Recognizing this pattern is the first crucial step. In the exercise, the expression \(64x^3 - 125\) can be seen as a difference of cubes because:

• \(64x^3\) is the cube of \(4x\)
• \(125\) is the cube of \(5\)

The algebraic formula for the difference of cubes is \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). By identifying \(a\) and \(b\), we can apply this formula to factorize the expression. This approach simplifies the seemingly complex polynomial into manageable parts.

Polynomial Expressions

**Polynomial expressions** are a fundamental component of algebra. They consist of terms that include variables raised to whole-number exponents. The polynomial given in the exercise, \(64x^3 - 125\), is an example with only two terms. This specific kind of polynomial is often called a binomial.
The challenge with polynomial expressions involves identifying recognizable patterns, such as cubes, squares, or other power relationships, that enable simplification through methods like factoring. By recognizing these patterns, particularly the form of cubes in this instance, one can transform the polynomial into a product of simpler expressions, facilitating easier manipulation and understanding.
Learners frequently practice with these expressions to develop proficiency in algebraic thinking and problem solving.

Algebraic Formulas

In algebra, **algebraic formulas** serve as critical tools for solving and simplifying expressions. The difference of cubes formula, mentioned earlier, is a prime example:

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

These formulas provide a blueprint for problem-solving, allowing us to systematically replace complex expressions with simple factored forms. In our exercise, identifying \(a = 4x\) and \(b = 5\) lets us apply the formula efficiently.
Using algebraic formulas not only makes calculations more manageable but also deepens the understanding of mathematical structures. They play a crucial role in higher-level math, equipping students with essential skills for advanced studies and applications in various scientific and engineering domains.