Factor The Trinomial 3x^2 + 17x + 10 Or Indicate If It Is A Prime Trinomial. Verify The Factorization Using The FOIL Method.

by ADMIN 125 views

Factoring trinomials is a fundamental skill in algebra, and mastering it opens doors to solving various mathematical problems. In this comprehensive guide, we will delve into the process of factoring the trinomial 3x^2 + 17x + 10. We'll explore the steps involved, provide detailed explanations, and emphasize the importance of checking our work using the FOIL (First, Outer, Inner, Last) method. Whether you're a student grappling with algebra concepts or simply seeking a refresher, this article will equip you with the knowledge and confidence to tackle trinomial factorization effectively.

Understanding Trinomials

Before we dive into the specific trinomial, let's first understand what trinomials are. A trinomial is a polynomial expression consisting of three terms. These terms typically involve a variable raised to different powers, along with constant coefficients. The general form of a quadratic trinomial is ax^2 + bx + c, where 'a', 'b', and 'c' are constants, and 'x' is the variable. In our case, the trinomial is 3x^2 + 17x + 10, where a = 3, b = 17, and c = 10. Factoring a trinomial means expressing it as a product of two binomials, which are polynomials with two terms each. This process is essentially the reverse of the FOIL method, which we'll use later to check our factorization.

The Factoring Process

Now, let's break down the process of factoring the trinomial 3x^2 + 17x + 10. There are several methods for factoring trinomials, but we'll focus on the method that involves finding two numbers that satisfy specific conditions related to the coefficients of the trinomial. Here are the steps:

Step 1: Identify the Coefficients

The first step is to identify the coefficients 'a', 'b', and 'c' in the trinomial. As we mentioned earlier, in our trinomial 3x^2 + 17x + 10, we have a = 3, b = 17, and c = 10. These coefficients play a crucial role in the factoring process.

Step 2: Find Two Numbers

This is the core of the factoring process. We need to find two numbers, let's call them 'm' and 'n', that satisfy two conditions:

  1. Their product is equal to the product of 'a' and 'c' (m * n = a * c).
  2. Their sum is equal to 'b' (m + n = b).

In our case, we need to find two numbers whose product is 3 * 10 = 30 and whose sum is 17. Let's list the pairs of factors of 30:

  • 1 and 30
  • 2 and 15
  • 3 and 10
  • 5 and 6

Out of these pairs, 2 and 15 satisfy both conditions: 2 * 15 = 30 and 2 + 15 = 17. So, m = 2 and n = 15.

Step 3: Rewrite the Middle Term

Now, we rewrite the middle term (17x) of the trinomial as the sum of two terms using the numbers we found in the previous step. In our case, we rewrite 17x as 2x + 15x. So, the trinomial becomes:

3x^2 + 2x + 15x + 10

Step 4: Factor by Grouping

Next, we factor by grouping. We group the first two terms and the last two terms together:

(3x^2 + 2x) + (15x + 10)

Now, we factor out the greatest common factor (GCF) from each group. From the first group, the GCF is 'x', and from the second group, the GCF is 5:

x(3x + 2) + 5(3x + 2)

Notice that we now have a common binomial factor (3x + 2) in both terms. We factor out this common binomial factor:

(3x + 2)(x + 5)

Step 5: The Factored Form

We have now successfully factored the trinomial. The factored form of 3x^2 + 17x + 10 is (3x + 2)(x + 5).

Checking the Factorization Using FOIL

To ensure that our factorization is correct, we can use the FOIL method to multiply the binomials (3x + 2) and (x + 5) and see if we get back the original trinomial. The FOIL method stands for:

  • First: Multiply the first terms of each binomial.
  • Outer: Multiply the outer terms of the binomials.
  • Inner: Multiply the inner terms of the binomials.
  • Last: Multiply the last terms of each binomial.

Let's apply the FOIL method to (3x + 2)(x + 5):

  • First: 3x * x = 3x^2
  • Outer: 3x * 5 = 15x
  • Inner: 2 * x = 2x
  • Last: 2 * 5 = 10

Now, add these products together:

3x^2 + 15x + 2x + 10

Combine the like terms (15x and 2x):

3x^2 + 17x + 10

This is the original trinomial, which confirms that our factorization is correct.

Why Factoring is Important

Factoring trinomials is a critical skill in algebra for several reasons:

  • Solving Equations: Factoring allows us to solve quadratic equations, which are equations of the form ax^2 + bx + c = 0. By factoring the quadratic expression, we can find the values of 'x' that make the equation true.
  • Simplifying Expressions: Factoring can simplify complex algebraic expressions, making them easier to work with. This is especially useful in calculus and other advanced math courses.
  • Graphing Functions: Factoring helps us find the x-intercepts of quadratic functions, which are the points where the graph of the function crosses the x-axis. This information is essential for graphing quadratic functions accurately.
  • Real-World Applications: Quadratic equations and factoring techniques have numerous applications in real-world scenarios, such as physics, engineering, and economics.

Common Mistakes to Avoid

While factoring trinomials might seem straightforward, there are some common mistakes that students often make. Being aware of these mistakes can help you avoid them:

  • Incorrectly Identifying Coefficients: Make sure you correctly identify the coefficients 'a', 'b', and 'c' in the trinomial. A mistake here can lead to incorrect factorization.
  • Finding the Wrong Numbers: The key to factoring trinomials is finding the two numbers 'm' and 'n' that satisfy the product and sum conditions. Double-check your calculations to ensure you've found the correct numbers.
  • Incorrectly Factoring by Grouping: When factoring by grouping, make sure you factor out the GCF correctly from each group. Also, ensure that the binomial factors in both terms are the same; otherwise, the grouping is incorrect.
  • Forgetting to Check: Always check your factorization using the FOIL method. This is the best way to catch any errors and ensure that your factorization is correct.

Practice Makes Perfect

Like any mathematical skill, mastering trinomial factorization requires practice. The more you practice, the more comfortable and confident you'll become with the process. Start with simple trinomials and gradually move on to more complex ones. Work through various examples, and don't hesitate to seek help from your teacher, classmates, or online resources if you get stuck.

Conclusion

Factoring the trinomial 3x^2 + 17x + 10 involves a systematic approach of identifying coefficients, finding the right numbers, rewriting the middle term, factoring by grouping, and, most importantly, checking your work using the FOIL method. This comprehensive guide has equipped you with the knowledge and steps to confidently factor trinomials. Remember, factoring is not just a mathematical exercise; it's a powerful tool with applications in various fields. So, keep practicing, and you'll soon find yourself effortlessly factoring trinomials and tackling more advanced algebraic problems.

By understanding the underlying principles and practicing diligently, you can master this essential skill and excel in your algebra journey. Happy factoring!