Factoring The Trinomial 9x² + 30x + 25 A Step-by-Step Guide

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Factoring trinomials is a fundamental skill in algebra, and mastering it opens doors to solving more complex equations and understanding polynomial behavior. In this comprehensive guide, we will delve into the process of factoring the specific trinomial 9x² + 30x + 25. This trinomial is a perfect square trinomial, a special type that exhibits a distinct pattern, making it easier to factor. We'll break down the steps, explain the underlying concepts, and provide you with a clear understanding of how to approach similar problems.

Recognizing Perfect Square Trinomials

Before diving into the factoring process, it's crucial to recognize what a perfect square trinomial is. A perfect square trinomial is a trinomial that results from squaring a binomial. It follows a specific pattern:

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

Notice the key characteristics: The first and last terms (a² and b²) are perfect squares, and the middle term (2ab) is twice the product of the square roots of the first and last terms. When you encounter a trinomial, checking if it fits this pattern is the first step in determining if it's a perfect square trinomial. In the given trinomial, 9x² + 30x + 25, we can see that 9x² is the square of 3x, and 25 is the square of 5. Now, we need to check if the middle term, 30x, fits the pattern. Is it twice the product of 3x and 5? Let's see: 2 * (3x) * 5 = 30x. Indeed, it matches! This confirms that 9x² + 30x + 25 is a perfect square trinomial, and we can proceed with factoring it accordingly.

Step-by-Step Factoring of 9x² + 30x + 25

Now that we've identified our trinomial as a perfect square, let's factor it step-by-step:

  1. Identify 'a' and 'b': In our trinomial, 9x² + 30x + 25, we recognize that 9x² is the square of 3x (so, a = 3x) and 25 is the square of 5 (so, b = 5). Understanding this correspondence is crucial for applying the perfect square trinomial pattern.
  2. Determine the Sign: The middle term, 30x, is positive. This tells us that the binomial we're looking for will have a '+' sign. If the middle term were negative, we'd use a '-' sign. This is because the expansion of (a + b)² results in positive cross-terms, while the expansion of (a - b)² results in negative cross-terms.
  3. Construct the Binomial: Based on our identified 'a', 'b', and the sign, we can construct the binomial: (3x + 5). This is the core of our factored expression. We've essentially found the square root of the perfect square trinomial.
  4. Write the Factored Form: Since we have a perfect square trinomial, the factored form is the square of the binomial we just constructed. Therefore, the factored form of 9x² + 30x + 25 is (3x + 5)². This means (3x + 5) multiplied by itself will give us the original trinomial.

Verification: Expanding the Factored Form

To ensure our factoring is correct, we can expand the factored form (3x + 5)² and see if it matches the original trinomial. Expanding (3x + 5)² means multiplying (3x + 5) by itself:

(3x + 5)² = (3x + 5)(3x + 5)

Using the distributive property (or the FOIL method), we multiply each term in the first binomial by each term in the second binomial:

  • 3x * 3x = 9x²
  • 3x * 5 = 15x
  • 5 * 3x = 15x
  • 5 * 5 = 25

Now, we combine these terms:

9x² + 15x + 15x + 25

Combining like terms (15x + 15x), we get:

9x² + 30x + 25

This is exactly the original trinomial we started with! This verification confirms that our factoring is correct. Expanding the factored form is a crucial step in ensuring the accuracy of your solution.

Alternative Method: The AC Method

While recognizing the perfect square trinomial pattern is the most efficient way to factor 9x² + 30x + 25, let's explore an alternative method called the AC method. This method is applicable to a broader range of trinomials, not just perfect squares. It involves breaking down the middle term into two terms, allowing us to factor by grouping. Here's how the AC method works for our example:

  1. Identify a, b, and c: In the trinomial 9x² + 30x + 25, a = 9, b = 30, and c = 25. These coefficients are the foundation of the AC method.
  2. Calculate AC: Multiply the coefficients 'a' and 'c': 9 * 25 = 225. This product, AC, is a key value in the method. We need to find two factors of 225 that add up to the middle coefficient, 'b' (which is 30).
  3. Find the Factors: We need to find two numbers that multiply to 225 and add up to 30. Through trial and error or by listing factors, we find that 15 and 15 satisfy these conditions (15 * 15 = 225 and 15 + 15 = 30). These factors will help us break down the middle term.
  4. Rewrite the Trinomial: Rewrite the middle term (30x) using the two factors we found: 15x and 15x. So, the trinomial becomes 9x² + 15x + 15x + 25. We've essentially split the middle term into two parts that will facilitate factoring by grouping.
  5. Factor by Grouping: Now, we group the first two terms and the last two terms: (9x² + 15x) + (15x + 25). We then factor out the greatest common factor (GCF) from each group.
    • From (9x² + 15x), the GCF is 3x. Factoring out 3x, we get 3x(3x + 5).
    • From (15x + 25), the GCF is 5. Factoring out 5, we get 5(3x + 5).
    • Now we have: 3x(3x + 5) + 5(3x + 5).
  6. Factor out the Common Binomial: Notice that both terms now have a common binomial factor: (3x + 5). We factor this out: (3x + 5)(3x + 5).
  7. Write the Factored Form: This simplifies to (3x + 5)², which is the same factored form we obtained using the perfect square trinomial pattern. The AC method, although longer, provides a systematic approach that works for various trinomials.

Common Mistakes to Avoid

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

  • Forgetting the Sign: Pay close attention to the signs of the terms in the trinomial. A wrong sign can lead to an incorrect factored form. Remember, the sign of the middle term in a perfect square trinomial indicates the sign in the binomial.
  • Incorrectly Identifying 'a' and 'b': Make sure you correctly identify the square roots of the first and last terms. For example, the square root of 9x² is 3x, not 9x or x².
  • Not Checking the Middle Term: Before assuming a trinomial is a perfect square, always verify that the middle term is twice the product of the square roots of the first and last terms. This is a crucial step in the process.
  • Incomplete Factoring: Always check if the factored binomial can be factored further. For example, if you end up with (2x + 4), you can factor out a 2 to get 2(x + 2).
  • Skipping Verification: Expanding the factored form is a vital step to ensure accuracy. Don't skip it! It's a simple way to catch errors.

Practice Problems

To solidify your understanding of factoring perfect square trinomials, try factoring these examples:

  1. 4x² + 20x + 25
  2. 16x² - 24x + 9
  3. 25x² + 60x + 36
  4. 49x² - 14x + 1
  5. x² + 8x + 16

By working through these problems, you'll gain confidence and proficiency in recognizing and factoring perfect square trinomials. Remember to follow the steps we've outlined, pay attention to the signs, and always verify your answers.

Conclusion

Factoring the trinomial 9x² + 30x + 25 demonstrates the elegance and efficiency of recognizing perfect square trinomials. By identifying the pattern and following the steps outlined, you can quickly and accurately factor this type of trinomial. Whether you use the perfect square trinomial pattern or the AC method, the key is to understand the underlying principles and practice regularly. Mastering factoring is a crucial step in your algebraic journey, and with dedication, you'll become proficient at it. Remember to always verify your answers and be mindful of common mistakes. Happy factoring!