Factoring The Trinomial 12r^2 + 29r + 15 Step By Step Guide

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Factoring trinomials is a fundamental skill in algebra, essential for simplifying expressions, solving equations, and understanding various mathematical concepts. In this article, we will delve into the process of factoring the trinomial 12r² + 29r + 15 step-by-step. We will explore the techniques involved, provide clear explanations, and offer helpful tips to master this skill. Whether you are a student learning algebra or someone looking to refresh your knowledge, this guide will equip you with the tools to confidently factor trinomials.

Understanding Trinomials and Factoring

Before we dive into the specifics of factoring 12r² + 29r + 15, 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. It generally takes the form of ax² + bx + c, where a, b, and c are constants, and x is a variable. In our case, the trinomial is 12r² + 29r + 15, where a = 12, b = 29, and c = 15. Factoring, in essence, is the reverse process of expansion. It involves breaking down a polynomial into a product of simpler expressions, typically binomials. These binomials, when multiplied together, yield the original trinomial. Factoring is a crucial skill in algebra for several reasons:

  • Simplifying Expressions: Factoring allows us to simplify complex polynomial expressions into more manageable forms. This is particularly useful when dealing with fractions or rational expressions, where factoring the numerator and denominator can reveal common factors that can be canceled out.
  • Solving Equations: Factoring is a key technique for solving quadratic equations (equations of the form ax² + bx + c = 0). By factoring the quadratic expression into two binomials, we can set each binomial equal to zero and solve for the variable. This method provides a direct and efficient way to find the roots or solutions of the equation.
  • Understanding Mathematical Concepts: Factoring provides a deeper understanding of the relationships between polynomials and their factors. It helps visualize how polynomials are constructed and how their roots are related to their coefficients. This conceptual understanding is invaluable for tackling more advanced algebraic problems.

Factoring trinomials might seem daunting at first, but with practice and a systematic approach, it becomes a manageable and even enjoyable process. The key is to understand the underlying principles and apply the appropriate techniques.

Step-by-Step Factoring of 12r² + 29r + 15

Now, let's embark on the journey of factoring the trinomial 12r² + 29r + 15. We'll break down the process into clear, manageable steps, providing explanations and insights along the way.

Step 1: Identify the Coefficients

The first step in factoring a trinomial is to identify the coefficients a, b, and c. In our trinomial, 12r² + 29r + 15, we have:

  • a = 12 (the coefficient of the r² term)
  • b = 29 (the coefficient of the r term)
  • c = 15 (the constant term)

These coefficients are the building blocks for our factoring process. They will guide us in finding the appropriate factors that will lead to the factored form of the trinomial.

Step 2: Calculate ac

The next step is to calculate the product of the coefficients a and c. This product, denoted as ac, plays a crucial role in finding the right combination of factors. In our case, ac = 12 * 15 = 180.

The value of ac provides us with a target number that we need to decompose into factors that will help us split the middle term (the term with coefficient b).

Step 3: Find Factors of ac that Add up to b

This is the heart of the factoring process. We need to find two numbers that satisfy two conditions:

  1. Their product is equal to ac (which is 180 in our case).
  2. Their sum is equal to b (which is 29 in our case).

Finding these two numbers might require some trial and error, but a systematic approach can make it easier. We can start by listing out the factor pairs of 180 and checking if their sum equals 29:

  • 1 and 180 (sum = 181)
  • 2 and 90 (sum = 92)
  • 3 and 60 (sum = 63)
  • 4 and 45 (sum = 49)
  • 5 and 36 (sum = 41)
  • 6 and 30 (sum = 36)
  • 9 and 20 (sum = 29) (This is our pair!)
  • 10 and 18 (sum = 28)
  • 12 and 15 (sum = 27)

We have found our pair! The numbers 9 and 20 satisfy both conditions: their product is 180, and their sum is 29. These numbers will be instrumental in splitting the middle term.

Step 4: Split the Middle Term

Now that we have found the two numbers, 9 and 20, we can split the middle term (29r) into two terms using these numbers as coefficients. We rewrite the trinomial 12r² + 29r + 15 as:

12r² + 9r + 20r + 15

Notice that we have simply replaced 29r with 9r + 20r. The value of the expression remains the same, but we have created a four-term expression that can be factored by grouping.

Step 5: Factor by Grouping

Factoring by grouping involves grouping the first two terms and the last two terms and then factoring out the greatest common factor (GCF) from each group. Let's apply this to our expression:

12r² + 9r + 20r + 15

Group the terms:

(12r² + 9r) + (20r + 15)

Factor out the GCF from each group:

  • The GCF of 12r² and 9r is 3r. Factoring out 3r, we get: 3r(4r + 3)
  • The GCF of 20r and 15 is 5. Factoring out 5, we get: 5(4r + 3)

Now our expression looks like this:

3r(4r + 3) + 5(4r + 3)

Notice that we have a common binomial factor of (4r + 3) in both terms. This is a crucial step in factoring by grouping. If we don't have a common binomial factor at this point, it indicates an error in our previous steps, and we need to go back and check our work.

Step 6: Factor out the Common Binomial

Since we have a common binomial factor of (4r + 3), we can factor it out from the entire expression:

3r(4r + 3) + 5(4r + 3) = (4r + 3)(3r + 5)

We have successfully factored out the common binomial, and we now have the factored form of the trinomial.

Step 7: Write the Factored Form

The final step is to write the factored form of the trinomial. We have found that:

12r² + 29r + 15 = (4r + 3)(3r + 5)

This is the factored form of the trinomial. We have expressed the trinomial as a product of two binomials.

Verification and Conclusion

To ensure that our factoring is correct, we can multiply the binomials back together and see if we get the original trinomial:

(4r + 3)(3r + 5) = 4r * 3r + 4r * 5 + 3 * 3r + 3 * 5

= 12r² + 20r + 9r + 15

= 12r² + 29r + 15

Our result matches the original trinomial, so we can confidently conclude that our factoring is correct.

In this comprehensive guide, we have meticulously factored the trinomial 12r² + 29r + 15. We have explored the underlying principles of factoring, broken down the process into manageable steps, and provided clear explanations and insights along the way. By mastering these techniques, you can confidently tackle factoring trinomials and apply this skill to various algebraic problems.

Additional Tips and Considerations for Factoring Trinomials

Factoring trinomials can sometimes be challenging, but with practice and a few additional tips, you can improve your skills and accuracy. Here are some important considerations to keep in mind:

  • Check for a Greatest Common Factor (GCF) First: Before attempting to factor a trinomial using the techniques described above, always check if there is a GCF that can be factored out from all three terms. Factoring out the GCF first simplifies the trinomial and makes the subsequent factoring process easier. For example, if we had the trinomial 24r² + 58r + 30, we could factor out a GCF of 2, resulting in 2(12r² + 29r + 15). We would then factor the trinomial 12r² + 29r + 15 as we did before and include the GCF in the final factored form.
  • Recognize Special Cases: There are certain types of trinomials that have specific factoring patterns. Recognizing these patterns can save you time and effort. Two common special cases are:
    • Perfect Square Trinomials: These trinomials can be factored into the square of a binomial. They have the form a² + 2ab + b² or a² - 2ab + b². For example, the trinomial 9r² + 12r + 4 is a perfect square trinomial because it can be factored as (3r + 2)².
    • Difference of Squares: While not technically a trinomial, the difference of squares pattern (a² - b²) is closely related to factoring. Recognizing this pattern can help you factor expressions that might initially seem complex. For example, the expression 4r² - 9 can be factored as (2r + 3)(2r - 3).
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