Factoring 24r² - 46r + 10 Completely A Step-by-Step Guide

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Factoring trinomials is a fundamental skill in algebra, and mastering it opens doors to solving quadratic equations, simplifying expressions, and tackling more advanced mathematical concepts. In this comprehensive guide, we'll delve into the process of factoring the trinomial 24r² - 46r + 10 completely. We'll break down each step, providing clear explanations and examples to ensure a solid understanding. Whether you're a student looking to improve your algebra skills or simply someone interested in revisiting mathematical concepts, this guide will equip you with the knowledge and confidence to factor trinomials effectively.

1. Identifying the Greatest Common Factor (GCF)

When factoring any polynomial, the first and most crucial step is to identify the greatest common factor (GCF) of all the terms. The GCF is the largest factor that divides evenly into each term of the polynomial. In the case of our trinomial, 24r² - 46r + 10, we need to find the GCF of 24, 46, and 10. To do this, we can list the factors of each number:

  • Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
  • Factors of 46: 1, 2, 23, 46
  • Factors of 10: 1, 2, 5, 10

By examining the lists, we can see that the greatest common factor of 24, 46, and 10 is 2. Since there is no common variable factor in all three terms (only the first two terms have 'r'), the GCF is simply 2. Now, we factor out the GCF from the trinomial:

24r² - 46r + 10 = 2(12r² - 23r + 5)

Factoring out the GCF simplifies the trinomial, making it easier to factor further. This step is essential because it reduces the size of the numbers we need to work with, leading to a more manageable factoring process. Always remember to look for the GCF first, as it can significantly simplify the problem. In this case, factoring out 2 leaves us with the trinomial 12r² - 23r + 5, which we will now focus on factoring completely.

2. Factoring the Simplified Trinomial (12r² - 23r + 5)

After extracting the GCF, we are left with the trinomial 12r² - 23r + 5. This is a quadratic trinomial in the form of ax² + bx + c, where a = 12, b = -23, and c = 5. There are several methods to factor such trinomials, but one common and effective approach is the AC method (also known as the grouping method). This method involves the following steps:

  1. Multiply a and c: Multiply the coefficient of the squared term (a) by the constant term (c). In our case, 12 * 5 = 60.

  2. Find two factors of ac that add up to b: We need to find two factors of 60 that add up to -23. Since the product is positive (60) and the sum is negative (-23), both factors must be negative. Let's list the pairs of factors of 60:

    • -1 and -60 (sum = -61)
    • -2 and -30 (sum = -32)
    • -3 and -20 (sum = -23) (This is the pair we need!)
    • -4 and -15 (sum = -19)
    • -5 and -12 (sum = -17)
    • -6 and -10 (sum = -16)

  3. Rewrite the middle term: Replace the middle term (-23r) with the two factors we found (-3r and -20r). So, the trinomial becomes:

    12r² - 23r + 5 = 12r² - 3r - 20r + 5

    This step is crucial because it sets up the trinomial for factoring by grouping.

  4. Factor by grouping: Group the first two terms and the last two terms, and then factor out the GCF from each group:

    (12r² - 3r) + (-20r + 5)

    From the first group (12r² - 3r), the GCF is 3r. Factoring it out, we get:

    3r(4r - 1)

    From the second group (-20r + 5), the GCF is -5. Factoring it out, we get:

    -5(4r - 1)

    Now, our expression looks like this:

    3r(4r - 1) - 5(4r - 1)

    Notice that both terms have a common factor of (4r - 1).

  5. Factor out the common binomial factor: Factor out the common binomial factor (4r - 1) from the entire expression:

    (4r - 1)(3r - 5)

    Thus, the factored form of 12r² - 23r + 5 is (4r - 1)(3r - 5). This process, while involving several steps, is a systematic way to factor trinomials of this form. The key is to find the correct factors of ac that add up to b, and then carefully apply the grouping method.

3. Combining the GCF with the Factored Trinomial

Having factored the simplified trinomial 12r² - 23r + 5 into (4r - 1)(3r - 5), we must now remember the GCF we factored out in the first step. The original trinomial was 24r² - 46r + 10, and we factored out a GCF of 2. To get the complete factored form, we need to include this GCF in our final answer. This means multiplying the factored trinomial by 2:

2(12r² - 23r + 5) = 2(4r - 1)(3r - 5)

Therefore, the completely factored form of the trinomial 24r² - 46r + 10 is 2(4r - 1)(3r - 5). This step is crucial to ensure we have factored the original trinomial completely, not just the simplified version. Forgetting to include the GCF is a common mistake, so always double-check that you have included it in your final answer.

4. Verification and Checking the Solution

After factoring a trinomial, it is always a good practice to verify the solution. This ensures that we have factored correctly and haven't made any mistakes along the way. There are two primary methods to verify our factored form: expansion and substitution.

4.1. Verification by Expansion

The most direct method to verify our factoring is to expand the factored form and see if it matches the original trinomial. We start with our factored form, 2(4r - 1)(3r - 5), and expand the binomial factors first:

(4r - 1)(3r - 5)

Using the FOIL method (First, Outer, Inner, Last), we multiply the binomials:

  • First: 4r * 3r = 12r²
  • Outer: 4r * -5 = -20r
  • Inner: -1 * 3r = -3r
  • Last: -1 * -5 = 5

Combining these terms, we get:

12r² - 20r - 3r + 5 = 12r² - 23r + 5

Now, we multiply the result by the GCF, which is 2:

2(12r² - 23r + 5) = 24r² - 46r + 10

This matches our original trinomial, 24r² - 46r + 10, which confirms that our factoring is correct. Expansion is a reliable method because it reverses the factoring process, allowing us to directly compare our result with the original expression.

4.2. Verification by Substitution

Another method to verify the factoring is by substitution. This involves choosing a value for the variable (in this case, 'r') and substituting it into both the original trinomial and the factored form. If the values are the same, it supports the correctness of our factoring. Let's choose a simple value, such as r = 1. Substitute r = 1 into the original trinomial:

24r² - 46r + 10 = 24(1)² - 46(1) + 10 = 24 - 46 + 10 = -12

Now, substitute r = 1 into the factored form:

2(4r - 1)(3r - 5) = 2(4(1) - 1)(3(1) - 5) = 2(4 - 1)(3 - 5) = 2(3)(-2) = -12

Both the original trinomial and the factored form evaluate to -12 when r = 1, which further supports that our factoring is correct. While one substitution is often sufficient, using multiple values can provide even greater confidence in the solution. However, it's important to choose values that are easy to calculate to avoid errors during substitution.

5. Common Mistakes to Avoid When Factoring Trinomials

Factoring trinomials can be challenging, and it's easy to make mistakes if you're not careful. Here are some common errors to avoid:

  1. Forgetting to factor out the GCF: This is one of the most frequent mistakes. Always look for the greatest common factor first and factor it out. If you skip this step, the remaining trinomial may be harder to factor, or you might miss a factor in your final answer.
  2. Incorrectly identifying factors: When using the AC method, ensure you find the correct factors of ac that add up to b. A small mistake in identifying factors can lead to incorrect factoring.
  3. Sign errors: Pay close attention to the signs of the factors and the terms in the trinomial. A sign error can completely change the factored form.
  4. Incorrectly grouping terms: When factoring by grouping, make sure you group the terms correctly and factor out the GCF from each group carefully. A mistake in grouping can lead to an incorrect result.
  5. Not checking the answer: Always verify your factored form by expanding it or using substitution. This will help you catch any mistakes and ensure that your solution is correct.
  6. Stopping too early: Ensure you have factored the trinomial completely. This means that each factor should be prime and cannot be factored further.
  7. Mixing up factoring methods: There are different methods for factoring trinomials, such as the AC method, trial and error, and special patterns. Make sure you choose the appropriate method for the given trinomial and apply it correctly.

By being aware of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in factoring trinomials. Practice and attention to detail are key to mastering this skill.

Conclusion

Factoring the trinomial 24r² - 46r + 10 completely involves a systematic approach that includes identifying the GCF, factoring the simplified trinomial, and verifying the solution. We found that the completely factored form is 2(4r - 1)(3r - 5). By following the steps outlined in this guide, you can confidently factor various trinomials. Remember to always look for the GCF first, apply the appropriate factoring method, and verify your answer to avoid common mistakes. With practice, factoring trinomials will become a manageable and rewarding skill in your mathematical journey.