Factoring Expressions Using The Greatest Common Factor A Step-by-Step Guide

Factoring expressions is a fundamental skill in algebra, allowing us to simplify complex expressions and solve equations. One of the first techniques to master is factoring out the greatest common factor (GCF). In this article, we'll explore how to factor expressions by identifying and extracting the GCF, using the example expression: $30y^{12} + 35y^7 - 25y^3$. This method helps break down expressions into simpler, more manageable forms, which is crucial for further algebraic manipulations and problem-solving.

Understanding the Greatest Common Factor (GCF)

At the heart of factoring lies the concept of the greatest common factor (GCF). The GCF is the largest factor that divides two or more numbers or terms without leaving a remainder. When factoring expressions, the GCF is the term with the highest degree and coefficient that can be divided evenly from all terms in the expression. Identifying the GCF is the first and most crucial step in the factoring process. To understand this better, let's break it down into simpler terms. Imagine you have a basket of fruits – some apples, some oranges, and some bananas. The GCF is like finding the largest group of fruits that you can take out, where each group has the same number of each type of fruit. In mathematical terms, this means finding the largest number and the highest power of a variable that can divide all terms in the expression without leaving any remainders. For instance, consider the numbers 12 and 18. The factors of 12 are 1, 2, 3, 4, 6, and 12, while the factors of 18 are 1, 2, 3, 6, 9, and 18. The greatest common factor between 12 and 18 is 6, as it is the largest number that appears in both lists of factors. Similarly, when dealing with algebraic expressions, we look for the highest common numerical factor and the highest common power of the variable. Understanding and identifying the GCF is not just a preliminary step; it's the backbone of factoring, making the entire process simpler and more efficient. Factoring out the GCF correctly sets the stage for further simplification and solving of algebraic problems, ensuring accuracy and saving time in the long run. Therefore, mastering this concept is essential for anyone looking to excel in algebra and beyond.

Identifying the GCF in $30y^{12} + 35y^7 - 25y^3$

To factor the expression $30y^{12} + 35y^7 - 25y^3$, the first critical step is to pinpoint the greatest common factor (GCF). This involves examining both the numerical coefficients and the variable terms within the expression. Let’s begin by breaking down the coefficients: 30, 35, and -25. We need to find the largest number that divides all three coefficients evenly. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The factors of 35 are 1, 5, 7, and 35. The factors of -25 are -1, -5, -25, 1, 5, and 25. By comparing these factors, we can see that the greatest common numerical factor is 5. This means that 5 is the largest number that can divide 30, 35, and -25 without leaving a remainder. Next, we turn our attention to the variable terms: $y^{12}$, $y^7$, and $y^3$. When determining the GCF for variables, we look for the lowest exponent present in the terms. This is because the lowest exponent represents the highest power of the variable that can be factored out from all terms. In this case, we have $y^{12}$, $y^7$, and $y^3$. The exponents are 12, 7, and 3, respectively. The smallest exponent is 3, so the greatest common variable factor is $y^3$. This means that $y^3$ is the highest power of y that is common to all terms. Now that we have identified both the greatest common numerical factor (5) and the greatest common variable factor ($y^3$), we can combine them to find the overall GCF of the expression. Multiplying these two factors together, we get $5y^3$. This is the GCF for the entire expression $30y^{12} + 35y^7 - 25y^3$. Understanding how to systematically identify the GCF is crucial because it sets the stage for the next step in factoring, which involves extracting this GCF from the expression. This foundational step ensures that the expression is simplified correctly, making subsequent algebraic manipulations more straightforward.

Factoring Out the GCF

Having identified the greatest common factor (GCF) as $5y^3$ in the expression $30y^{12} + 35y^7 - 25y^3$, the next step is to factor it out. Factoring out the GCF involves dividing each term in the expression by the GCF and writing the expression as a product of the GCF and the resulting quotient. This process effectively reverses the distributive property, allowing us to simplify the expression into a more manageable form. To begin, we divide each term in the original expression by $5y^3$. This means we'll perform the following divisions: $30y^{12} / 5y^3$, $35y^7 / 5y^3$, and $-25y^3 / 5y^3$. Let's break down each division step by step. First, divide $30y^{12}$ by $5y^3$. Dividing the coefficients, we get $30 / 5 = 6$. When dividing variables with exponents, we subtract the exponents: $y^{12} / y^3 = y^{(12-3)} = y^9$. So, the result of the first division is $6y^9$. Next, we divide $35y^7$ by $5y^3$. Again, we divide the coefficients: $35 / 5 = 7$. For the variables, we subtract the exponents: $y^7 / y^3 = y^{(7-3)} = y^4$. Thus, the result of the second division is $7y^4$. Finally, we divide $-25y^3$ by $5y^3$. Dividing the coefficients, we get $-25 / 5 = -5$. For the variables, we subtract the exponents: $y^3 / y^3 = y^{(3-3)} = y^0$. Since any variable raised to the power of 0 is 1, $y^0 = 1$. Therefore, the result of the third division is -5. Now that we have performed all the divisions, we can write the factored expression. We write the GCF, $5y^3$, outside the parentheses and the results of our divisions inside the parentheses. This gives us: $5y^3(6y^9 + 7y^4 - 5)$. This factored form represents the original expression in a simplified manner, where the GCF is clearly separated from the remaining terms. This is a crucial step in simplifying algebraic expressions and solving equations, as it allows us to work with smaller, more manageable terms. Factoring out the GCF not only simplifies the expression but also makes it easier to identify further factoring opportunities or to solve equations where this expression is set equal to zero. The ability to correctly factor out the GCF is a fundamental skill in algebra, paving the way for more complex algebraic manipulations.

Final Factored Form

After successfully identifying the greatest common factor (GCF) as $5y^3$ and factoring it out of the expression $30y^{12} + 35y^7 - 25y^3$, we arrive at the final factored form: $5y^3(6y^9 + 7y^4 - 5)$. This representation is crucial because it simplifies the original expression, making it easier to analyze, manipulate, and use in further algebraic operations. The final factored form clearly shows the GCF, $5y^3$, as a standalone factor, multiplied by the expression inside the parentheses, $6y^9 + 7y^4 - 5$. This form not only simplifies the expression visually but also reveals key information about its structure. For instance, it becomes easier to identify the roots of the expression if it were part of an equation set equal to zero. The factored form allows us to apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. In this case, setting $5y^3(6y^9 + 7y^4 - 5) = 0$ gives us two potential cases: $5y^3 = 0$ or $6y^9 + 7y^4 - 5 = 0$. The first case, $5y^3 = 0$, immediately tells us that $y = 0$ is a solution. The second case, $6y^9 + 7y^4 - 5 = 0$, is a polynomial equation of higher degree, which may require more advanced techniques to solve. However, having the expression in factored form makes it clear that solving this equation is necessary to find all the roots of the original expression. Moreover, the factored form can be useful in simplifying rational expressions, adding or subtracting fractions with polynomial denominators, and other algebraic manipulations. It provides a clearer picture of the components of the expression and their relationships, which is essential for advanced problem-solving. In summary, the final factored form $5y^3(6y^9 + 7y^4 - 5)$ is not just an end result but also a stepping stone to further mathematical analysis and application. It encapsulates the simplification achieved by factoring out the GCF, highlighting the importance of this technique in algebra and beyond.

Factoring out the greatest common factor is a fundamental technique in algebra. By identifying and extracting the GCF, we simplify complex expressions, making them easier to work with in various mathematical contexts. The final factored form, $5y^3(6y^9 + 7y^4 - 5)$, represents a simplified version of the original expression, highlighting the power and utility of factoring in algebraic manipulations.