Factoring The Trinomial: $15x^2 - 26x + 8$ - A Step-by-Step Guide

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Factoring trinomials can seem daunting, but with a systematic approach, even complex expressions like $15x^2 - 26x + 8$ can be broken down into simpler factors. In this comprehensive guide, we will walk through the process step-by-step, ensuring you understand each stage and can apply the techniques to other trinomials. This process involves identifying the coefficients, applying the AC method, grouping terms, and extracting common factors. By the end of this guide, you'll have a solid understanding of how to factor this specific trinomial and similar expressions.

Understanding Trinomial Factoring

Before diving into the specifics of $15x^2 - 26x + 8$, let's first understand what trinomial factoring entails. A trinomial is a polynomial with three terms, typically in the form of $ax^2 + bx + c$, where a, b, and c are constants. Factoring a trinomial means expressing it as a product of two binomials (expressions with two terms). The ability to factor trinomials is a fundamental skill in algebra, essential for solving quadratic equations, simplifying expressions, and tackling more advanced mathematical problems.

Factoring trinomials is a crucial skill in algebra because it allows us to simplify complex expressions, solve equations, and understand the behavior of polynomials. When we factor a trinomial, we're essentially reversing the process of multiplication. Think of it like this: multiplication combines two expressions into one, while factoring breaks one expression into two or more simpler expressions. This process is particularly useful when dealing with quadratic equations, which often appear in various mathematical and real-world contexts. Mastering trinomial factoring not only helps in solving mathematical problems but also enhances problem-solving skills in general, enabling you to approach complex tasks with a structured methodology.

The trinomial $15x^2 - 26x + 8$ fits this form, where $a = 15$, $b = -26$, and $c = 8$. The goal is to find two binomials $(px + q)$ and $(rx + s)$ such that their product equals the original trinomial. This involves a bit of algebraic detective work, where we identify the right combination of numbers that satisfy certain conditions derived from the coefficients of the trinomial. This process not only simplifies the expression but also reveals the roots of the corresponding quadratic equation, making it a powerful tool in algebraic manipulation and problem-solving.

Step 1: Identifying the Coefficients

The first step in factoring the trinomial $15x^2 - 26x + 8$ is to identify the coefficients $a$, $b$, and $c$. In this case:

• $a = 15$ (the coefficient of $x^2$)
• $b = -26$ (the coefficient of $x$)
• $c = 8$ (the constant term)

These coefficients play a crucial role in the factoring process. We'll use them to find two numbers that satisfy specific conditions, which will then help us break down the trinomial into its factors. Understanding the role of each coefficient is essential for applying the AC method effectively, which we'll discuss in the next step. Correctly identifying these values sets the foundation for the subsequent steps and ensures that the factoring process proceeds accurately.

The coefficients $a$, $b$, and $c$ dictate the structure of the trinomial and, consequently, the approach we take to factor it. The leading coefficient, $a$, influences the possible factors of the quadratic term, while the coefficient $b$ affects the middle term and the interaction between the two binomials. The constant term, $c$, provides crucial clues about the constants in the binomial factors. Together, these coefficients act as a blueprint for the factoring process. A keen understanding of their individual roles and how they interact with each other is key to mastering the art of factoring trinomials.

This initial identification is not just a preliminary step; it's the cornerstone of the entire factoring process. Any error in identifying these coefficients can lead to incorrect factors and a flawed solution. Therefore, taking a moment to double-check these values is always a good practice. It is this meticulous approach that transforms factoring from a daunting task into a manageable and even enjoyable puzzle.

Step 2: Applying the AC Method

The AC method is a powerful technique for factoring trinomials of the form $ax^2 + bx + c$. The first step in this method is to multiply the coefficients $a$ and $c$. In our case:

$a imes c = 15 imes 8 = 120$

Next, we need to find two numbers that multiply to 120 and add up to $b$ (which is -26). This is often the most challenging part, as it requires a bit of trial and error. We're looking for two numbers that, when multiplied, give us 120, and when added, give us -26. Since the product is positive and the sum is negative, we know that both numbers must be negative. The pairs of factors of 120 are:

1. -1 and -120
2. -2 and -60
3. -3 and -40
4. -4 and -30
5. -5 and -24
6. -6 and -20
7. -8 and -15
8. -10 and -12

Among these pairs, -6 and -20 satisfy both conditions:

• $-6 imes -20 = 120$
• $-6 + (-20) = -26$

Therefore, the two numbers we need are -6 and -20. The AC method hinges on this critical step of identifying the correct pair of numbers, as they will dictate how we rewrite the middle term of the trinomial in the next stage. This method's elegance lies in its ability to transform a complex trinomial into a more manageable expression that can be factored by grouping.

The success of the AC method largely depends on finding the right pair of numbers. This can sometimes involve listing out all the factor pairs of $ac$ and then checking their sums. It's a methodical process that rewards patience and attention to detail. The numbers we find here are the key to unlocking the trinomial's hidden structure, allowing us to break it down into its fundamental components. It’s like solving a puzzle where each piece (the factors) has to fit perfectly to reveal the final picture (the factored trinomial).

The AC method is not just a mathematical trick; it's a technique rooted in the fundamental properties of numbers and algebraic expressions. It highlights the relationship between multiplication and addition, and how they interplay in the structure of a trinomial. Mastering this method provides a deeper understanding of factoring, allowing you to approach various algebraic problems with confidence and skill.

Step 3: Rewriting the Middle Term

Now that we have identified -6 and -20 as the two numbers, we can rewrite the middle term (-26x) of the trinomial $15x^2 - 26x + 8$ using these numbers. This step is crucial because it sets up the trinomial for factoring by grouping. We replace -26x with -6x - 20x:

$15x^2 - 26x + 8 = 15x^2 - 6x - 20x + 8$

This rewriting doesn't change the value of the expression; it merely rearranges it in a way that allows us to factor by grouping. The choice of which number to write first (-6x or -20x) doesn't matter; the result will be the same. However, it's often helpful to consider which grouping might lead to simpler factoring in the next step. In this case, both arrangements are equally straightforward.

Rewriting the middle term is a strategic maneuver in the factoring process. It transforms a trinomial, which might seem difficult to factor directly, into a four-term polynomial that can be factored using a simple grouping technique. This step is a testament to the power of algebraic manipulation, demonstrating how rearranging terms can reveal hidden structures and simplify complex expressions. The elegance of this approach lies in its ability to leverage the distributive property in reverse, allowing us to break down the trinomial into more manageable components.

The careful selection of the numbers -6 and -20 is what makes this rewriting possible. These numbers, derived from the AC method, ensure that the resulting four-term polynomial can be neatly grouped and factored. This step is not just a mechanical manipulation; it's an insightful application of number theory and algebraic principles. It highlights the interconnectedness of different mathematical concepts and demonstrates how they can be used in tandem to solve problems.

Step 4: Factoring by Grouping

After rewriting the middle term, we now have a four-term polynomial: $15x^2 - 6x - 20x + 8$. Factoring by grouping involves pairing the terms and factoring out the greatest common factor (GCF) from each pair.

First, group the first two terms and the last two terms:

$(15x^2 - 6x) + (-20x + 8)$

Now, factor out the GCF from each group. The GCF of $15x^2$ and $-6x$ is $3x$, and the GCF of $-20x$ and $8$ is $-4$:

$3x(5x - 2) - 4(5x - 2)$

Notice that both terms now have a common binomial factor, $(5x - 2)$. This is a critical checkpoint in the factoring process. If the binomial factors are not the same, it indicates an error in the previous steps, and you may need to revisit your calculations.

Now, factor out the common binomial factor $(5x - 2)$:

$(5x - 2)(3x - 4)$

This is the factored form of the trinomial. Factoring by grouping is a powerful technique that transforms a four-term polynomial into a product of two binomials. It relies on the principle of distributing and factoring out common factors, revealing the underlying structure of the expression. This method is widely applicable and forms a cornerstone of algebraic manipulation.

The beauty of factoring by grouping lies in its systematic approach. By carefully pairing terms and extracting common factors, we gradually simplify the expression until it reaches its factored form. This step-by-step process minimizes the chances of error and provides a clear path to the solution. The common binomial factor that emerges during this process is a testament to the correctness of the preceding steps, serving as a confirmation that we are on the right track.

Factoring by grouping is not just a technical procedure; it's an exercise in pattern recognition and algebraic thinking. It encourages us to look for commonalities and connections within the expression, revealing the inherent order and structure. This skill is invaluable in various mathematical contexts, from solving equations to simplifying complex expressions. Mastering this technique not only enhances your algebraic proficiency but also sharpens your problem-solving abilities in general.

Step 5: Verification

To verify that our factored form is correct, we can multiply the two binomials back together and see if we get the original trinomial. This step is crucial to ensure that no errors were made during the factoring process. It's like checking your work in any problem-solving scenario, giving you confidence in your solution.

Multiply $(5x - 2)(3x - 4)$ using the distributive property (often remembered by the acronym FOIL - First, Outer, Inner, Last):

• First: $5x imes 3x = 15x^2$
• Outer: $5x imes -4 = -20x$
• Inner: $-2 imes 3x = -6x$
• Last: $-2 imes -4 = 8$

Now, combine the terms:

$15x^2 - 20x - 6x + 8 = 15x^2 - 26x + 8$

This matches the original trinomial, so our factored form is correct. Verification is an essential part of the factoring process. It not only confirms the correctness of the solution but also reinforces the understanding of the relationship between factors and products. This step is particularly important when dealing with complex expressions, where the chances of making a mistake are higher.

Verification is not just a formality; it's an opportunity to deepen your understanding of factoring. By multiplying the factors back together, you're essentially reversing the factoring process, gaining a clearer picture of how the trinomial is constructed. This dual perspective – breaking down the expression and then building it back up – is invaluable for developing a robust understanding of algebraic concepts.

This final check brings closure to the problem-solving process, instilling confidence in your abilities and reinforcing the importance of accuracy in mathematical work. It's a testament to the power of careful, methodical problem-solving and the satisfaction of arriving at a correct solution through a clear, logical process.

Conclusion

The trinomial $15x^2 - 26x + 8$ can be factored into $(5x - 2)(3x - 4)$. We achieved this by following a systematic approach: identifying the coefficients, applying the AC method, rewriting the middle term, factoring by grouping, and finally, verifying our solution. This step-by-step guide provides a clear and concise method for factoring trinomials, a fundamental skill in algebra. Mastering this process not only enables you to solve specific problems but also enhances your overall mathematical understanding and problem-solving abilities. Factoring trinomials is a journey that blends algebraic manipulation with numerical reasoning. It's a skill that, once mastered, unlocks a deeper understanding of mathematical structures and relationships.

This process illustrates the power of systematic problem-solving in mathematics. By breaking down a complex problem into smaller, more manageable steps, we can navigate even the most challenging expressions. The AC method, combined with factoring by grouping, provides a robust framework for factoring trinomials of this form. The verification step adds an extra layer of assurance, confirming the correctness of our solution and reinforcing our understanding of the process.

Factoring is not just a mathematical technique; it's a way of thinking, a method of approaching complex problems with clarity and precision. It's a skill that transcends the classroom, finding applications in various fields, from engineering to computer science. The ability to break down complex problems into simpler components is a valuable asset in any domain, and factoring trinomials is an excellent training ground for developing this skill. We encourage you to continue practicing factoring with various trinomials to solidify your understanding and build confidence in your algebraic abilities. For more information and practice, check out this helpful resource on factoring from Khan Academy.