Problem 84 - Beginning and Intermediate Algebra

Question

Factor completely. $$64 c^{3}+1$$

Step-by-Step Solution

Answer

The factored form of the given expression is \((4c + 1)(16c^2 - 4c + 1)\).

Step 1: Rewrite the expression as sum of cubes

First, we need to rewrite the expression as a sum of cubes: \(64c^3 + 1 = (4c)^3 + 1^3\)

Step 2: Applying the sum of cubes formula

The sum of cubes formula is: \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). In our case, \(a = 4c\) and \(b = 1\). Apply the formula to the expression: \((4c)^3 + (1)^3 = (4c + 1)((4c)^2 - (4c)(1) + (1)^2)\)

Step 3: Expand and simplify

Now, we expand and simplify: \((4c + 1)(16c^2 - 4c + 1)\) So the factored form of the given expression is \((4c + 1)(16c^2 - 4c + 1)\).

Key Concepts

Sum of Cubes

The concept of the "sum of cubes" is a fundamental topic in algebra. It involves expressing a number or an expression as a sum of two cubed terms. The standard mathematical representation for the sum of cubes is shown as:

• \(a^3 + b^3\)

In our exercise, we are tasked with factoring \(64c^3 + 1\).
By recognizing that \(64c^3\) is \((4c)^3\) and \(1\) is \((1)^3\), we see that the problem can be rewritten using the sum of cubes formula. Thus, it becomes \((4c)^3 + (1)^3\).
This allows us to utilize specific algebraic formulas to split and manipulate the expression further. Understanding how to break down expressions into cubes is essential, as it simplifies solving complex polynomial equations in algebra.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra. They consist of variables, constants, and arithmetic operations. In essence, an algebraic expression is a combination of numbers and letters representing real-world quantities. The expression \(64c^3 + 1\) involves these elements:

• Coefficient \(64\): Multiplies the variable \(c^3\).
• Variable \(c\): Represents an unknown value, raised to the third power \(c^3\).
• Constant \(1\): An unchanging number added to the term.

This expression is simplified by recognizing patterns like the sum of cubes, which facilitates further algebraic manipulation. Understanding how to identify parts of expressions enables you to apply appropriate formulas or rules to solve, simplify, or factor them. Thus, algebraic expressions are crucial in representing mathematical ideas in a structured way.

Factoring Techniques

Factoring techniques are strategies used to decompose algebraic expressions into products of simpler expressions. Different patterns and formulas are used based on the expression's structure. The sum of cubes follows a specific formula:

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

For our example:

• Set \(a = 4c\) and \(b = 1\).
• Applying the formula, \((4c)^3 + (1)^3\) becomes \((4c + 1)((4c)^2 - 4c \cdot 1 + 1^2)\).
• Expand to get: \((4c + 1)(16c^2 - 4c + 1)\).

Factoring is powerful for simplifying expressions, solving equations, and understanding their properties. The ability to factor efficiently allows one to solve polynomial equations and gain insights into their solutions and roots, making it a vital skill in mathematics.