Factoring The Trinomial M^2 + 12m + 35 A Step-by-Step Guide

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In the realm of algebra, factoring trinomials is a fundamental skill. It's the process of breaking down a trinomial into the product of two binomials. This skill is crucial for solving quadratic equations, simplifying algebraic expressions, and understanding the behavior of polynomial functions. In this comprehensive guide, we will delve into the process of factoring the trinomial m^2 + 12m + 35. We'll explore the underlying principles, the steps involved, and provide examples to solidify your understanding.

Understanding Trinomials and Factoring

Before we dive into the specifics of factoring m^2 + 12m + 35, let's establish a solid foundation. A trinomial is a polynomial with three terms. In the general form ax^2 + bx + c, where a, b, and c are constants, and x is the variable. Factoring, in essence, is the reverse of the distributive property (or the FOIL method). When we factor a trinomial, we're looking for two binomials that, when multiplied together, give us the original trinomial.

In our case, the trinomial is m^2 + 12m + 35. Notice that the coefficient of the m^2 term (a) is 1. This makes the factoring process slightly simpler. Our goal is to find two binomials of the form (m + p) and (m + q), where p and q are constants, such that:

(m + p)(m + q) = m^2 + 12m + 35

The Factoring Process: Finding the Right Numbers

The key to factoring a trinomial like m^2 + 12m + 35 lies in finding the right pair of numbers. These numbers, p and q, must satisfy two conditions:

  1. Their product must equal the constant term (c): In our case, p * q = 35.
  2. Their sum must equal the coefficient of the middle term (b): In our case, p + q = 12.

Let's systematically find these numbers. We need to identify the factors of 35. The factors of 35 are 1 and 35, as well as 5 and 7. Now, we need to check which pair of factors adds up to 12.

  • 1 + 35 = 36 (This pair doesn't work)
  • 5 + 7 = 12 (This pair works!)

We've found our numbers! p = 5 and q = 7. This means our binomial factors will be (m + 5) and (m + 7).

Putting It Together: The Factored Form

Now that we've identified the numbers, we can write the factored form of the trinomial:

m^2 + 12m + 35 = (m + 5)(m + 7)

This is the solution! We have successfully factored the trinomial into the product of two binomials.

Verifying the Solution: Expanding the Binomials

It's always a good practice to verify your factoring by expanding the binomials using the distributive property (or the FOIL method). Let's expand (m + 5)(m + 7):

(m + 5)(m + 7) = m(m + 7) + 5(m + 7)

  • = m^2 + 7m + 5m + 35
  • = m^2 + 12m + 35

As you can see, expanding the binomials gives us back the original trinomial, m^2 + 12m + 35. This confirms that our factoring is correct.

Examples and Practice Problems

To further solidify your understanding, let's look at a few more examples and practice problems.

Example 1: Factor x^2 + 8x + 15

  1. Find the factors of 15: The factors of 15 are 1 and 15, as well as 3 and 5.
  2. Identify the pair that adds up to 8: 3 + 5 = 8
  3. Write the factored form: (x + 3)(x + 5)

Example 2: Factor y^2 + 10y + 24

  1. Find the factors of 24: The factors of 24 are 1 and 24, 2 and 12, 3 and 8, and 4 and 6.
  2. Identify the pair that adds up to 10: 4 + 6 = 10
  3. Write the factored form: (y + 4)(y + 6)

Practice Problems:

  1. Factor a^2 + 9a + 20
  2. Factor b^2 + 11b + 28
  3. Factor c^2 + 13c + 42

(Answers: 1. (a + 4)(a + 5), 2. (b + 4)(b + 7), 3. (c + 6)(c + 7))

Common Mistakes to Avoid

Factoring trinomials can be tricky, and there are some common mistakes to watch out for:

  • Incorrectly identifying factors: Make sure you list all the factor pairs of the constant term and carefully check their sums.
  • Mixing up signs: Pay close attention to the signs of the terms in the trinomial. If the constant term is positive and the middle term is negative, both factors will be negative. If the constant term is negative, one factor will be positive, and one will be negative.
  • Forgetting to verify: Always verify your factoring by expanding the binomials to ensure you get back the original trinomial.

Factoring Trinomials with a Leading Coefficient (a > 1)

While we've focused on trinomials where the coefficient of the squared term (a) is 1, the factoring process becomes slightly more complex when a is greater than 1. In these cases, we often use a method called the "ac method" or factoring by grouping.

Let's briefly outline the steps involved in the ac method:

  1. Multiply a and c: Calculate the product of the leading coefficient and the constant term.
  2. Find factors of ac that add up to b: Identify two numbers whose product is ac and whose sum is the coefficient of the middle term (b).
  3. Rewrite the middle term: Replace the middle term (bx) with the sum of two terms using the factors found in step 2.
  4. Factor by grouping: Group the terms in pairs and factor out the greatest common factor (GCF) from each pair.
  5. Write the factored form: If done correctly, you should have a common binomial factor that can be factored out, resulting in the final factored form.

Factoring trinomials with a leading coefficient greater than 1 requires more steps and careful attention to detail. It's essential to practice these types of problems to develop proficiency.

Conclusion

Factoring the trinomial m^2 + 12m + 35 is a straightforward process that involves finding two numbers that multiply to the constant term and add up to the coefficient of the middle term. By following the steps outlined in this guide, you can confidently factor trinomials of this form. Remember to verify your solutions by expanding the binomials and to practice regularly to master this essential algebraic skill. Understanding factoring trinomials is crucial for solving more complex algebraic problems and for a deeper understanding of mathematics. With consistent practice factoring and attention to detail, you can develop the skills needed to factor algebraic expressions effectively. This guide on factoring provides a solid foundation for understanding trinomial factorization and its applications in algebraic problem-solving. Keep practicing, and you'll become a factoring pro in no time! The factors of m^2 + 12m + 35 are (m + 5) and (m + 7). These factors represent the binomial expressions that, when multiplied together, result in the original trinomial. This process of finding the factors is a fundamental concept in algebra and is essential for simplifying expressions and solving equations. By mastering this skill, you'll be well-equipped to tackle more advanced mathematical concepts. The ability to factor trinomials is a cornerstone of algebraic proficiency and is crucial for success in higher-level mathematics courses. So, continue to practice and hone your skills in trinomial factoring to build a strong foundation in algebra. This comprehensive guide has provided you with the knowledge and tools necessary to factor expressions like m^2 + 12m + 35 with confidence and accuracy. Remember to always double-check your work and seek out additional practice problems to reinforce your understanding. With dedication and perseverance, you'll become a master of factoring and unlock the door to a deeper understanding of mathematics.