Factoring Trinomials A Step By Step Guide To Factoring 15x^2 + 32x - 7
Factoring trinomials is a fundamental skill in algebra, essential for solving quadratic equations, simplifying expressions, and understanding polynomial behavior. In this comprehensive guide, we will delve into the process of completely factoring the trinomial 15x² + 32x - 7. We'll break down the steps, explore the underlying principles, and provide examples to solidify your understanding. Whether you're a student learning algebra for the first time or someone looking to refresh your skills, this guide will equip you with the knowledge and techniques to confidently factor trinomials.
Understanding Trinomials and Factoring
Before we dive into the specific trinomial, let's establish a clear understanding of what trinomials are and why factoring is important. A trinomial is a polynomial expression consisting of three terms. The general form of a quadratic trinomial is ax² + bx + c, where a, b, and c are constants, and x is the variable. In our case, the trinomial 15x² + 32x - 7 fits this form, with a = 15, b = 32, and c = -7.
Factoring is the process of breaking down a polynomial expression into a product of simpler expressions. In the context of trinomials, factoring involves finding two binomials that, when multiplied together, yield the original trinomial. Factoring is crucial because it allows us to solve quadratic equations. When a quadratic equation is factored, we can set each factor equal to zero and solve for x, thus finding the roots or solutions of the equation. Factoring also simplifies algebraic expressions, making them easier to work with in further calculations or manipulations.
Strategies for Factoring Trinomials
Several strategies can be employed to factor trinomials, each with its strengths and applications. Here, we will focus on the AC method, a widely used and effective technique, especially for trinomials with a leading coefficient (the coefficient of the x² term) that is not 1. The AC method provides a systematic approach, reducing the trial and error often associated with factoring. By following a clear set of steps, we can efficiently break down complex trinomials into their binomial factors.
The AC method hinges on manipulating the trinomial in a way that allows us to factor by grouping, a technique that involves identifying common factors within pairs of terms. By understanding the logic behind the AC method and practicing its application, you will gain a powerful tool for tackling a wide range of trinomial factoring problems. This method not only helps in factoring but also enhances your understanding of the relationship between the coefficients of the trinomial and its factors.
Applying the AC Method to 15x² + 32x - 7
Now, let's apply the AC method to factor the trinomial 15x² + 32x - 7 step-by-step. This will illustrate the practical application of the method and highlight its effectiveness. Each step is crucial, and understanding the reasoning behind each step will make the process clearer and more intuitive.
Step 1: Identify a, b, and c
The first step is to identify the coefficients a, b, and c in the trinomial. In 15x² + 32x - 7, we have:
- a = 15
- b = 32
- c = -7
Identifying these coefficients is the foundation for the AC method, as they are used in subsequent calculations. Correctly identifying a, b, and c ensures that the factoring process proceeds accurately.
Step 2: Calculate AC
Next, we calculate the product of a and c, denoted as AC:
AC = a * c = 15 * (-7) = -105
This product, AC, is a crucial value that guides the rest of the factoring process. It helps us find the pair of numbers that will allow us to rewrite the middle term of the trinomial.
Step 3: Find Two Numbers That Multiply to AC and Add to B
This is the heart of the AC method. We need to find two numbers that multiply to AC (-105) and add up to b (32). This step often involves some trial and error, but with practice, you'll become adept at identifying these pairs. We are essentially looking for factors of -105 that have a difference of 32. Think systematically about the factors of 105 and their combinations. Prime factorization can be a helpful tool here.
After considering various possibilities, we find that the numbers 35 and -3 satisfy these conditions:
- 35 * (-3) = -105
- 35 + (-3) = 32
Finding this pair of numbers is the key to rewriting the trinomial in a way that allows us to factor by grouping. These numbers essentially "split" the middle term, setting the stage for the next step.
Step 4: Rewrite the Middle Term
Using the numbers we found (35 and -3), we rewrite the middle term (32x) of the trinomial as the sum of two terms:
15x² + 32x - 7 = 15x² + 35x - 3x - 7
This step transforms the trinomial into a four-term expression, which is amenable to factoring by grouping. Notice that we have simply replaced 32x with 35x - 3x, without changing the overall value of the expression.
Step 5: Factor by Grouping
Now, we factor by grouping, which involves pairing the first two terms and the last two terms and finding the greatest common factor (GCF) in each pair:
- From the first pair (15x² + 35x), the GCF is 5x. Factoring out 5x, we get: 5x(3x + 7)
- From the second pair (-3x - 7), the GCF is -1. Factoring out -1, we get: -1(3x + 7)
Notice that factoring out a negative GCF from the second pair is crucial in ensuring that the binomial factors match. This alignment is necessary for the final factoring step.
Step 6: Factor Out the Common Binomial
We now have two terms: 5x(3x + 7) and -1(3x + 7). Observe that both terms have a common binomial factor: (3x + 7). We factor out this common binomial:
5x(3x + 7) - 1(3x + 7) = (3x + 7)(5x - 1)
This step completes the factoring process. We have successfully broken down the trinomial into a product of two binomials.
The Factored Form
Therefore, the completely factored form of the trinomial 15x² + 32x - 7 is:
(3x + 7)(5x - 1)
We have successfully factored the trinomial using the AC method. This factored form can now be used to solve quadratic equations, simplify algebraic expressions, and gain further insights into the behavior of the trinomial.
Verifying the Result
It's always a good practice to verify your factoring result. This can be done by multiplying the binomial factors back together and ensuring that the result matches the original trinomial. This step provides a check for errors and reinforces your understanding of the factoring process.
To verify, we multiply (3x + 7) and (5x - 1) using the distributive property (often remembered by the acronym FOIL - First, Outer, Inner, Last):
(3x + 7)(5x - 1) = (3x * 5x) + (3x * -1) + (7 * 5x) + (7 * -1)
= 15x² - 3x + 35x - 7
= 15x² + 32x - 7
Since the result matches the original trinomial, we can confidently conclude that our factoring is correct. This verification step underscores the importance of accuracy and thoroughness in algebraic manipulations.
Conclusion
In this comprehensive guide, we have successfully factored the trinomial 15x² + 32x - 7 using the AC method. We explored the fundamental concepts of trinomials and factoring, the importance of factoring in algebra, and the step-by-step application of the AC method. Each step, from identifying coefficients to verifying the result, plays a crucial role in the process.
Factoring trinomials is a skill that improves with practice. By working through various examples and applying different techniques, you can develop your proficiency and confidence in this area of algebra. Remember, the AC method provides a systematic approach, but understanding the underlying principles allows you to adapt and apply it effectively in different situations. With dedication and practice, you'll master the art of factoring trinomials and unlock a deeper understanding of algebraic expressions and equations.
This skill will be invaluable as you progress further in mathematics, whether you're tackling more advanced algebraic concepts or applying these principles in real-world problem-solving scenarios. The ability to factor trinomials efficiently and accurately will serve as a solid foundation for your mathematical journey.
