Factoring The Trinomial 10r² + 19r - 15 A Step-by-Step Guide

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Factoring trinomials is a fundamental skill in algebra, and mastering it opens doors to solving various mathematical problems. In this article, we will delve into the process of factoring the trinomial 10r² + 19r - 15 using a systematic approach. We will explore the steps involved, the underlying principles, and provide a clear, concise explanation that will help you understand and apply this technique to other similar problems. Whether you're a student grappling with algebra or someone looking to brush up on your math skills, this guide will provide you with the tools and knowledge you need to confidently factor trinomials.

Understanding Trinomial Factoring

Before we dive into the specific trinomial, let's first understand the concept of factoring trinomials. A trinomial is a polynomial expression with three terms. Factoring a trinomial involves expressing it as a product of two binomials. This process is essentially the reverse of multiplying two binomials using the FOIL (First, Outer, Inner, Last) method. The goal is to find two binomials that, when multiplied together, give you the original trinomial.

When faced with factoring a trinomial like 10r² + 19r - 15, the first step is to identify the coefficients of the terms. In this case, we have a quadratic term (10r²), a linear term (19r), and a constant term (-15). The coefficients are 10, 19, and -15, respectively. These coefficients play a crucial role in determining the factors of the trinomial. The factoring process often involves trial and error, but with a systematic approach and a good understanding of the relationships between the coefficients, you can significantly reduce the number of attempts needed.

One common technique for factoring trinomials is the ac method. This method involves finding two numbers that multiply to the product of the leading coefficient (a) and the constant term (c), and add up to the middle coefficient (b). In our case, a = 10, b = 19, and c = -15. So, we need to find two numbers that multiply to (10)(-15) = -150 and add up to 19. This is a critical step, and finding the correct numbers is key to successfully factoring the trinomial. Once these numbers are found, they are used to rewrite the middle term of the trinomial, which then allows us to factor by grouping.

Applying the ac Method to 10r² + 19r - 15

Now, let's apply the ac method to our trinomial, 10r² + 19r - 15. As mentioned earlier, we need to find two numbers that multiply to -150 (10 * -15) and add up to 19. This might seem daunting at first, but we can systematically search for these numbers by considering the factors of -150. We need one positive and one negative factor since the product is negative. Let's list some factor pairs of 150:

  • 1 and 150
  • 2 and 75
  • 3 and 50
  • 5 and 30
  • 6 and 25
  • 10 and 15

Now, we need to consider the negative counterparts and see which pair has a difference of 19. After examining the pairs, we find that 25 and -6 satisfy the conditions. 25 multiplied by -6 equals -150, and 25 plus -6 equals 19. These are the numbers we need.

Next, we rewrite the middle term (19r) using these two numbers. So, 19r becomes 25r - 6r. Our trinomial now looks like this: 10r² + 25r - 6r - 15. This step is crucial because it sets us up to factor by grouping, a technique that simplifies the factoring process by breaking it down into smaller, more manageable parts. By rewriting the middle term, we have effectively transformed the trinomial into a four-term expression, which is easier to factor by grouping.

Factoring by grouping involves pairing the terms and factoring out the greatest common factor (GCF) from each pair. In our expression, we pair the first two terms and the last two terms: (10r² + 25r) + (-6r - 15). Now, we find the GCF of each pair. The GCF of 10r² and 25r is 5r, and the GCF of -6r and -15 is -3. Factoring out these GCFs, we get: 5r(2r + 5) - 3(2r + 5). Notice that both terms now have a common binomial factor, which is (2r + 5). This is a key indicator that we are on the right track.

Completing the Factorization

Now that we have the common binomial factor (2r + 5), we can factor it out from the entire expression. This gives us: (2r + 5)(5r - 3). This is the factored form of the trinomial 10r² + 19r - 15. We have successfully expressed the trinomial as a product of two binomials.

To verify our result, we can multiply the two binomials using the FOIL method:

  • First: (2r)(5r) = 10r²
  • Outer: (2r)(-3) = -6r
  • Inner: (5)(5r) = 25r
  • Last: (5)(-3) = -15

Adding these terms together, we get: 10r² - 6r + 25r - 15 = 10r² + 19r - 15, which is the original trinomial. This confirms that our factorization is correct.

Therefore, the factored form of the trinomial 10r² + 19r - 15 is (2r + 5)(5r - 3). This process demonstrates the power of the ac method and factoring by grouping in simplifying quadratic expressions. By following these steps, you can confidently factor a wide range of trinomials.

In summary, factoring the trinomial 10r² + 19r - 15 involves the following key steps:

  1. Identify the coefficients a, b, and c.
  2. Find two numbers that multiply to ac and add up to b.
  3. Rewrite the middle term using these two numbers.
  4. Factor by grouping.
  5. Verify the result by multiplying the binomials.

By mastering these steps, you can confidently tackle trinomial factoring problems and enhance your algebraic skills. Factoring trinomials is not just a mathematical exercise; it's a crucial skill that is applied in various fields, including engineering, physics, and computer science. A solid understanding of factoring will undoubtedly benefit you in your academic and professional pursuits.

Conclusion

In conclusion, we have successfully factored the trinomial 10r² + 19r - 15 into (2r + 5)(5r - 3) using the ac method and factoring by grouping. This process highlights the importance of a systematic approach and a strong understanding of algebraic principles. Factoring trinomials is a fundamental skill in mathematics, and by mastering it, you can unlock a deeper understanding of algebra and its applications. The steps outlined in this article provide a clear and concise guide to factoring trinomials, and with practice, you can confidently apply these techniques to solve a variety of problems. Remember, the key to success in factoring is to practice regularly and develop a keen eye for recognizing patterns and relationships between numbers. With dedication and the right tools, you can become proficient in factoring trinomials and excel in your mathematical endeavors.