Factoring Expressions A Comprehensive Guide To 24y^3 - 30y^2 + 9y

Jul 27, 2025 by James Vasile 66 views

Hey guys! Today, we are going to dive deep into the world of factoring expressions, specifically focusing on the expression $24y^3 - 30y^2 + 9y$ . Factoring is a crucial skill in algebra, and mastering it will help you solve a variety of mathematical problems. So, let’s get started and break down this expression step by step.

Understanding Factoring

Before we jump into the specifics, let’s quickly recap what factoring is all about. Factoring is the process of breaking down an expression into its constituent parts or factors. Think of it like reverse multiplication. For instance, if we multiply 3 and 4, we get 12. So, the factors of 12 are 3 and 4. Similarly, in algebra, we look for expressions that, when multiplied together, give us the original expression.

The most common type of factoring involves finding the Greatest Common Factor (GCF). The GCF is the largest factor that divides evenly into all terms of the expression. Identifying and factoring out the GCF is often the first step in simplifying and solving algebraic expressions. It makes the remaining expression easier to handle and reduces the complexity of the problem.

Factoring is not just a mathematical exercise; it has practical applications in various fields. In engineering, factoring helps simplify complex equations used in designing structures or systems. In computer science, it’s used in algorithm optimization and data compression. Even in finance, factoring can be used to simplify financial models and calculations. So, understanding how to factor expressions completely is a valuable skill that extends beyond the classroom.

Step 1: Identifying the Greatest Common Factor (GCF)

In the given expression, $24y^3 - 30y^2 + 9y$ , our first task is to identify the GCF. The GCF is the largest factor that divides evenly into all terms of the expression. To find it, we need to consider both the coefficients (the numbers) and the variables.

Let's start with the coefficients: 24, -30, and 9. We need to find the largest number that divides all three. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The factors of 9 are 1, 3, and 9. Looking at these, the largest number that appears in all three lists is 3. So, the numerical part of our GCF is 3.

Now, let’s consider the variables. We have $y^3$ , $y^2$ , and $y$ . Remember, when finding the GCF of variables, we take the lowest power of the common variable. In this case, the lowest power of y is $y^1$ , which is simply y. So, the variable part of our GCF is y.

Combining the numerical and variable parts, the GCF of the entire expression is 3y. This means that 3y is the largest term that can evenly divide each term in the expression $24y^3 - 30y^2 + 9y$ . Recognizing this GCF is the crucial first step, as it simplifies the expression and makes further factoring easier.

Understanding the GCF isn't just about finding a common number and variable; it’s about simplifying the expression to its most basic form. It’s like finding the common thread that runs through all the terms, allowing us to unravel the expression and see its underlying structure. By identifying the GCF, we set the stage for the next step in the factoring process.

Step 2: Factoring out the GCF

Now that we have identified the GCF as 3y, the next step is to factor it out from the expression $24y^3 - 30y^2 + 9y$ . Factoring out the GCF involves dividing each term in the expression by 3y and writing the result inside parentheses. It’s like distributing in reverse.

Let's break this down term by term:

Divide $24y^3$ by 3y: $\frac{24y^3}{3y} = 8y^2$
Divide $-30y^2$ by 3y: $\frac{-30y^2}{3y} = -10y$
Divide $9y$ by 3y: $\frac{9y}{3y} = 3$

So, when we divide each term by the GCF, we get $8y^2$ , $-10y$ , and 3. Now, we write these terms inside parentheses and place the GCF, 3y, outside the parentheses. This gives us:

$3y(8y^2 - 10y + 3)$

This is a crucial step because it simplifies the original expression significantly. By factoring out the GCF, we have reduced the complexity of the problem. The expression inside the parentheses is now a quadratic trinomial, which is easier to factor than the original cubic expression. Factoring out the GCF is like peeling away the outer layers to reveal the simpler core.

This process not only makes the expression more manageable but also sets the stage for further factoring if needed. The expression inside the parentheses might be factorable itself, which leads us to the next step. But for now, we’ve successfully factored out the GCF, making the problem much more approachable.

Step 3: Factoring the Quadratic Trinomial

After factoring out the GCF, we are left with the expression $3y(8y^2 - 10y + 3)$ . Our next task is to factor the quadratic trinomial inside the parentheses, which is $8y^2 - 10y + 3$ . To factor a quadratic trinomial in the form of $ax^2 + bx + c$ , we need to find two numbers that multiply to ac and add up to b.

In our case, a = 8, b = -10, and c = 3. So, we need to find two numbers that multiply to $8 * 3 = 24$ and add up to -10. Let’s list the factor pairs of 24 and see which pair adds up to -10:

1 and 24
2 and 12
3 and 8
4 and 6

Considering the negative signs, we can see that -4 and -6 are the numbers we need because $(-4) * (-6) = 24$ and $(-4) + (-6) = -10$ . Now, we rewrite the middle term (-10y) using these numbers:

$8y^2 - 10y + 3 = 8y^2 - 4y - 6y + 3$

Next, we factor by grouping. We group the first two terms and the last two terms:

$(8y^2 - 4y) + (-6y + 3)$

Now, we factor out the GCF from each group:

$4y(2y - 1) - 3(2y - 1)$

Notice that we now have a common factor of $(2y - 1)$ . We factor this out:

$(4y - 3)(2y - 1)$

So, the factored form of the quadratic trinomial $8y^2 - 10y + 3$ is $(4y - 3)(2y - 1)$ . This step involves a bit of algebraic manipulation, but it’s a fundamental technique for factoring quadratic expressions. By breaking down the trinomial and using the factoring by grouping method, we were able to express it as a product of two binomials.

Factoring the quadratic trinomial is like solving a puzzle. You need to find the right pieces that fit together to form the original expression. Each step, from finding the right numbers to factoring by grouping, is crucial in arriving at the final factored form.

Step 4: Writing the Completely Factored Expression

We have now factored out the GCF and factored the quadratic trinomial. It's time to combine our results and write the completely factored expression. We started with $24y^3 - 30y^2 + 9y$ and went through the following steps:

Identified the GCF as 3y.
Factored out the GCF: $3y(8y^2 - 10y + 3)$ .
Factored the quadratic trinomial $8y^2 - 10y + 3$ as $(4y - 3)(2y - 1)$ .

Now, we combine these results to get the completely factored expression. We write the GCF, 3y, followed by the factored form of the quadratic trinomial:

$3y(4y - 3)(2y - 1)$

This is the final, completely factored form of the expression $24y^3 - 30y^2 + 9y$ . We have broken it down into its simplest components. The expression is now represented as a product of three factors: 3y, (4y - 3), and (2y - 1).

Writing the completely factored expression is like putting the final touches on a masterpiece. Each step we took—identifying the GCF, factoring it out, and factoring the quadratic trinomial—was essential in achieving the final result. The completely factored expression not only simplifies the original expression but also reveals its structure and the relationships between its terms.

This factored form is incredibly useful for solving equations, simplifying fractions, and understanding the behavior of polynomial functions. It allows us to easily identify the roots of the equation, which are the values of y that make the expression equal to zero. In this case, the roots are y = 0, y = 3/4, and y = 1/2. The completely factored expression is a powerful tool for further mathematical analysis and problem-solving.

Conclusion

So, guys, we’ve successfully factored the expression $24y^3 - 30y^2 + 9y$ completely. We walked through the process step by step, from identifying the GCF to factoring the quadratic trinomial. Factoring is a crucial skill in algebra, and by mastering these techniques, you’ll be well-equipped to tackle more complex problems.

Remember, the key to successful factoring is practice. The more you practice, the more comfortable you’ll become with the different techniques and patterns. So, keep practicing, and you’ll become a factoring pro in no time! Factoring is not just about getting the right answer; it's about developing a deeper understanding of algebraic expressions and their properties. It’s a skill that will serve you well in many areas of mathematics and beyond.

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