Factoring $125a^3 + 1$: A Step-by-Step Guide

In this detailed guide, we will walk you through the process of factoring the expression $125a^3 + 1$ completely. This type of problem falls under the category of sum of cubes, a common pattern in algebra. Understanding how to factor expressions like this is crucial for simplifying equations, solving problems, and building a strong foundation in mathematics. Let’s break down the steps and master this concept!

Understanding the Sum of Cubes Formula

Before we dive into the specific problem, it’s essential to understand the general formula for the sum of cubes. The sum of cubes formula is a fundamental concept in algebra, and it states that:

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

This formula allows us to factor expressions that fit the pattern of a cube of one term plus the cube of another term. Recognizing this pattern is the first step in successfully factoring such expressions. The formula essentially breaks down the sum of two cubes into a product of a binomial $(a + b)$ and a trinomial $(a^2 - ab + b^2)$. Each component plays a crucial role in the factored form, and understanding their derivation helps in applying the formula effectively. Understanding this formula is key to tackling problems like factoring $125a^3 + 1$.

The sum of cubes formula is not just a mathematical trick; it's a powerful tool derived from the distributive property and algebraic manipulation. To truly appreciate its utility, let’s briefly explore its derivation. Imagine expanding the right side of the formula: $(a + b)(a^2 - ab + b^2)$. Using the distributive property, we multiply each term in the first parenthesis by each term in the second parenthesis:

$a(a^2 - ab + b^2) + b(a^2 - ab + b^2) = a^3 - a^2b + ab^2 + a^2b - ab^2 + b^3$

Notice how the terms $-a^2b$ and $+a^2b$ cancel each other out, as do the terms $+ab^2$ and $-ab^2$. This leaves us with:

$a^3 + b^3$

This derivation not only validates the sum of cubes formula but also provides insight into why it works. The careful arrangement of terms in the binomial and trinomial factors ensures that the intermediate terms cancel out, leaving only the sum of the cubes. This understanding can aid in memorization and application of the formula, as it connects the formula to fundamental algebraic principles.

Moreover, recognizing the sum of cubes pattern is crucial in various mathematical contexts, including simplifying algebraic expressions, solving equations, and even in calculus when dealing with limits and series. The ability to quickly identify and apply the sum of cubes formula can significantly streamline problem-solving and enhance mathematical fluency. Therefore, mastering this formula and its applications is an invaluable skill for anyone studying mathematics or related fields.

Step 1: Recognize the Pattern

The given expression is $125a^3 + 1$. We need to see if it fits the sum of cubes pattern, which is $a^3 + b^3$. To do this, we need to express both terms as perfect cubes.

• $125a^3$ can be written as $(5a)^3$ because $5^3 = 125$ and $(a)^3 = a^3$.
• $1$ can be written as $1^3$ since $1^3 = 1$.

Now we can clearly see that our expression fits the sum of cubes pattern, where:

• $a = 5a$
• $b = 1$

Recognizing this pattern is a crucial first step because it allows us to apply the sum of cubes formula directly. Without recognizing the pattern, we might struggle to find an appropriate factoring method. The ability to identify perfect cubes quickly is a valuable skill in algebra, as it unlocks a powerful factoring technique. Furthermore, this recognition extends beyond just numerical coefficients; it involves understanding how variables and their exponents interact within the expression. In this case, we not only recognize that 125 is a perfect cube (5 cubed) but also that $a^3$ is a perfect cube. This holistic understanding allows us to rewrite the expression in a form that readily aligns with the sum of cubes formula, paving the way for successful factorization.

In mathematical problem-solving, pattern recognition is often the key to unlocking solutions. It allows us to leverage established formulas and techniques, transforming complex problems into manageable steps. By identifying the sum of cubes pattern in $125a^3 + 1$, we've essentially mapped out our strategy for factoring the expression. This proactive approach is a hallmark of effective problem-solving in mathematics, where recognizing underlying structures can lead to elegant and efficient solutions.

Step 2: Apply the Sum of Cubes Formula

Now that we’ve identified $a = 5a$ and $b = 1$, we can substitute these values into the sum of cubes formula:

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

Substituting our values, we get:

$(5a)^3 + 1^3 = (5a + 1)((5a)^2 - (5a)(1) + 1^2)$

This step is a direct application of the formula, where we replace the generic variables $a$ and $b$ with their corresponding values from our specific expression. Accuracy in this step is crucial, as any error in substitution will lead to an incorrect factored form. The process highlights the importance of understanding the formula's structure and how each term corresponds to the original expression. By carefully substituting the values, we transform the abstract formula into a concrete representation of the factored expression. This methodical approach ensures that we maintain the integrity of the algebraic relationship and set the stage for further simplification.

The substitution process also underscores the elegance of algebraic formulas. By encapsulating a general pattern, these formulas allow us to efficiently handle a wide range of specific cases. In this instance, the sum of cubes formula provides a template for factoring any expression that fits the $a^3 + b^3$ pattern. This ability to generalize and apply abstract concepts to concrete situations is a fundamental aspect of mathematical thinking. It empowers us to solve problems systematically and efficiently, reducing complex tasks to a series of well-defined steps.

Step 3: Simplify the Expression

Next, we need to simplify the expression we obtained in the previous step:

$(5a + 1)((5a)^2 - (5a)(1) + 1^2)$

Let's simplify the terms inside the second parenthesis:

• $(5a)^2 = 25a^2$
• $(5a)(1) = 5a$
• $1^2 = 1$

Substituting these simplified terms back into the expression, we get:

$(5a + 1)(25a^2 - 5a + 1)$

This simplification step is crucial for obtaining the final factored form of the expression. It involves applying basic algebraic operations, such as squaring and multiplication, to reduce the terms within the parentheses. Each simplification must be performed accurately to ensure that the final factored form is correct. The process also demonstrates the importance of order of operations, as we first address the exponents and multiplications before combining the terms. By meticulously simplifying each term, we transform the expression into its most concise and understandable form.

Furthermore, the simplification process reinforces our understanding of algebraic notation and conventions. We correctly apply the power rule to $(5a)^2$, recognizing that the exponent applies to both the coefficient and the variable. We also understand that multiplying by 1 leaves the term unchanged. These seemingly small details are fundamental to algebraic manipulation and contribute to our overall mathematical proficiency. By mastering these basic operations, we build a solid foundation for tackling more complex algebraic problems.

Step 4: Check for Further Factoring

Now we have the expression $(5a + 1)(25a^2 - 5a + 1)$. We need to check if the quadratic expression $(25a^2 - 5a + 1)$ can be factored further.

To determine this, we can check its discriminant. The discriminant of a quadratic expression $ax^2 + bx + c$ is given by the formula:

$D = b^2 - 4ac$

In our case, $a = 25$, $b = -5$, and $c = 1$. So, the discriminant is:

$D = (-5)^2 - 4(25)(1) = 25 - 100 = -75$

Since the discriminant is negative, the quadratic expression has no real roots, which means it cannot be factored further using real numbers. This is a critical step because it ensures that we have completely factored the original expression. The discriminant provides valuable information about the nature of the roots of a quadratic equation, allowing us to determine whether it can be factored into linear factors with real coefficients. A negative discriminant indicates that the quadratic has complex roots, and therefore, cannot be factored further within the realm of real numbers.

The use of the discriminant also demonstrates the interconnectedness of different mathematical concepts. By applying the discriminant from quadratic equation theory, we can effectively analyze the factorability of the quadratic trinomial. This interdisciplinary approach is a hallmark of advanced mathematical thinking, where concepts from different areas of mathematics are combined to solve problems. Furthermore, understanding the discriminant and its implications is crucial for solving quadratic equations and analyzing their solutions. It provides a powerful tool for determining the nature and number of roots, and it is widely used in various applications of quadratic equations.

Final Answer

Therefore, the completely factored form of $125a^3 + 1$ is:

$(5a + 1)(25a^2 - 5a + 1)$

This is our final answer. We have successfully factored the expression using the sum of cubes formula and verified that the resulting quadratic trinomial cannot be factored further. This final factored form represents the most simplified representation of the original expression, highlighting its structure and components. The process of arriving at this answer involved a series of logical steps, from recognizing the pattern to applying the formula and simplifying the result. Each step contributed to the overall solution, demonstrating the importance of a methodical approach to mathematical problem-solving.

The factored form allows us to better understand the expression's behavior and properties. It can be used to solve equations, simplify expressions, and analyze functions. The ability to factor algebraic expressions is a fundamental skill in mathematics, with applications in various fields, including calculus, linear algebra, and beyond. By mastering these techniques, we equip ourselves with powerful tools for tackling a wide range of mathematical challenges.

In conclusion, factoring $125a^3 + 1$ completely requires recognizing the sum of cubes pattern, applying the appropriate formula, simplifying the resulting expression, and verifying that no further factoring is possible. This process not only yields the final factored form but also reinforces our understanding of algebraic principles and techniques. By practicing these types of problems, we can build our mathematical fluency and problem-solving skills.

For more information on factoring and algebraic identities, you can visit Khan Academy's Algebra resources.