Problem 93 - Beginning and Intermediate Algebra

Question

Factor completely. $$6 c^{3}+48$$

Step-by-Step Solution

Answer

The given expression \(6c^{3} + 48\) can be factored completely as \(6(c + 2)(c^{2} - 2c + 4)\).

Step 1: Determine the Greatest Common Factor (GCF)

First, we need to find the GCF of both terms in the expression \(6c^{3}\) and \(48\). The GCF of 6 and 48 is 6, so we can factor 6 out from the expression.

Step 2: Factor out the GCF

Factor out the GCF (6) from the given expression: \[6c^{3} + 48 = 6(c^{3} + 8)\]

Step 3: Identify Factoring Techniques

Now, we are left with the expression \(c^{3} + 8\). This is a sum of two cubes, which can be factored using the sum of cubes formula, where \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\). Here, the sum of cubes is \(c^{3} + 8 = c^{3} + 2^{3}\), so, we have \(a = c\) and \(b = 2\).

Step 4: Apply the Sum of Cubes Formula

Using the sum of cubes formula, we factor \(c^{3} + 8\) as: \[(c + 2)(c^{2} - 2c + 4)\]

Step 5: Write the Completely Factored Expression

Combining Steps 2 and 4, we have the completely factored expression: \[6c^{3} + 48 = 6(c + 2)(c^{2} - 2c + 4)\] Thus, the given expression \(6c^{3} + 48\) can be factored completely as \(6(c + 2)(c^{2} - 2c + 4)\).

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a crucial concept when it comes to simplifying polynomial expressions. The GCF is the largest number that evenly divides all terms in a polynomial expression. In the given exercise, we had the expression \(6c^3 + 48\). We looked at the coefficients of both terms: 6 and 48.

• 6 can be factored into prime numbers as \(2 \times 3\).
• 48 can be factored into \(2^4 \times 3\).

The GCF here is the product of the smallest powers of all common primes from the coefficients. In this case, both numbers could provide one \(2\) and one \(3\), so the GCF is \(6\).
By factoring out the GCF from the entire polynomial expression, we simplify it to the form \(6(c^3 + 8)\). This critical step reduces its complexity, preparing us for further factoring of the remaining polynomial \(c^3 + 8\).

Identifying the GCF simplifies calculations and reduces errors in solving polynomial expressions. It is a foundational skill in algebra.

Sum of Cubes Formula

The sum of cubes formula is a powerful tool for factoring specific polynomial expressions. A cube, in algebra, refers to raising a term to the power of three, say \(a^3\). When you have an expression that matches the form \(a^3 + b^3\), you can apply the sum of cubes formula:

• The formula states \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).

In our exercise, after factoring out the GCF, we were left with \(c^3 + 8\). Notice that 8 can be written as \(2^3\), making our expression \(c^3 + 2^3\).

We can identify \(a = c\) and \(b = 2\) and apply the sum of cubes formula. Substituting these values into the formula gives us \((c + 2)(c^2 - 2c + 4)\).
This step is vital as it transforms the polynomial into a product of two simpler expressions. It is much easier to handle smaller expressions in further algebraic manipulations or evaluations. Understanding and memorizing the sum of cubes formula can significantly simplify complex problem-solving scenarios.

Polynomial Expressions

Polynomial expressions are mathematical expressions consisting of variables, coefficients, and exponents. They are combined using addition, subtraction, multiplication, and non-negative integer exponents. In the exercise, we started with the polynomial \(6c^3 + 48\).

Polynomials can vary greatly in complexity:

• Monomials have only one term, like \(5x^2\).
• Binomials have two terms, e.g., \(3x + 4\).
• Trinomials include three terms, such as \(x^2 - x + 3\).
• Higher-degree polynomials can have multiple terms and powers, like the cubic polynomial \(c^3 + 8\).

Factoring polynomial expressions is a key algebraic skill since it allows us to simplify them and find solutions to polynomial equations. By breaking down polynomials into product form, we can solve equations more easily, graph curves, and study polynomial behavior at specific values.
In the example, we reduced a cubic polynomial to a product of a constant factor and simpler polynomials. This showcases how understanding the structure and factorization of polynomials is essential for advanced mathematical problem-solving. Mastery of polynomial manipulation is vital in fields ranging from engineering to economics.