Factoring The Perfect Square Trinomial 16x² + 40x + 25

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Factoring quadratic expressions is a fundamental skill in algebra. One particular type of quadratic expression, the perfect square trinomial, exhibits a special pattern that allows for quick and efficient factoring. In this article, we will delve into factoring the perfect square trinomial 16x² + 40x + 25 completely. We'll explore the characteristics of perfect square trinomials, the steps involved in factoring them, and how to verify the result. Mastering this technique not only simplifies algebraic manipulations but also lays the groundwork for solving quadratic equations and tackling more advanced mathematical concepts. So, let's embark on this journey to understand and conquer the art of factoring perfect square trinomials.

Understanding Perfect Square Trinomials

Before diving into the specific expression, 16x² + 40x + 25, let's first understand what defines a perfect square trinomial. A perfect square trinomial is a trinomial (a polynomial with three terms) that results from squaring a binomial (a polynomial with two terms). In other words, it can be expressed in one of two forms:

  • (a + b)² = a² + 2ab + b²
  • (a - b)² = a² - 2ab + b²

Key characteristics help us identify perfect square trinomials:

  1. The first and last terms are perfect squares: This means they can be written as the square of some expression. In our example, 16x² is the square of 4x (since (4x)² = 16x²), and 25 is the square of 5 (since 5² = 25).
  2. The middle term is twice the product of the square roots of the first and last terms: This is the crucial link that confirms the perfect square trinomial pattern. In 16x² + 40x + 25, the square root of 16x² is 4x, and the square root of 25 is 5. Twice their product is 2 * (4x) * 5 = 40x, which matches the middle term.

Recognizing these characteristics is the first step in efficiently factoring perfect square trinomials. By identifying the perfect square pattern, we can avoid lengthy trial-and-error methods and directly apply the appropriate factoring formula. This recognition not only saves time but also provides a deeper understanding of the structure of algebraic expressions. Now that we have a firm grasp of the definition, let's move on to the step-by-step process of factoring our specific trinomial.

Step-by-Step Factoring of 16x² + 40x + 25

Now that we know what a perfect square trinomial is, let's apply this knowledge to factor 16x² + 40x + 25. We'll follow a systematic approach, breaking down the process into manageable steps:

  1. Verify it's a perfect square trinomial: We've already done this in the previous section, but let's reiterate the key checks:

    • Is the first term (16x²) a perfect square? Yes, it's (4x)².
    • Is the last term (25) a perfect square? Yes, it's 5².
    • Is the middle term (40x) twice the product of the square roots of the first and last terms? Yes, 2 * (4x) * 5 = 40x. Since all conditions are met, we can confidently proceed with factoring as a perfect square trinomial.

  2. Identify 'a' and 'b': This is where we determine the expressions that are being squared. From the first term, 16x², we see that a = 4x. From the last term, 25, we see that b = 5.

  3. Determine the sign: The sign of the middle term tells us whether we'll use the (a + b)² or (a - b)² pattern. Since the middle term, 40x, is positive, we'll use the (a + b)² pattern.

  4. Apply the formula: We know that a² + 2ab + b² factors into (a + b)². Substituting our values for 'a' and 'b', we get: 16x² + 40x + 25 = (4x + 5)²

  5. Write the factored form: (4x + 5)² can also be written as (4x + 5)(4x + 5).

Therefore, the completely factored form of 16x² + 40x + 25 is (4x + 5)(4x + 5). This step-by-step process not only provides the solution but also reinforces the understanding of the underlying pattern. By meticulously checking each condition and applying the appropriate formula, we can confidently factor perfect square trinomials. In the next section, we'll further solidify our understanding by verifying the factored form to ensure accuracy.

Verifying the Factored Form

After factoring a polynomial, it's always a good practice to verify the result. This ensures that we haven't made any errors in the factoring process. To verify our factored form of (4x + 5)(4x + 5), we'll expand it and see if it matches the original expression, 16x² + 40x + 25. Expansion involves multiplying the binomials together, which can be done using the distributive property (often remembered by the acronym FOIL: First, Outer, Inner, Last).

Let's expand (4x + 5)(4x + 5):

  • First: (4x) * (4x) = 16x²
  • Outer: (4x) * (5) = 20x
  • Inner: (5) * (4x) = 20x
  • Last: (5) * (5) = 25

Now, combine the terms:

16x² + 20x + 20x + 25 = 16x² + 40x + 25

As we can see, the expanded form, 16x² + 40x + 25, matches our original trinomial. This confirms that our factored form, (4x + 5)(4x + 5), is indeed correct. Verification provides a crucial check in the factoring process, catching any potential mistakes and solidifying our confidence in the result. It also reinforces the connection between factoring and expanding, highlighting the inverse relationship between these two fundamental algebraic operations. With the factored form verified, we can confidently conclude our analysis of this perfect square trinomial.

Conclusion

In conclusion, we have successfully factored the perfect square trinomial 16x² + 40x + 25 completely. We began by understanding the characteristics of perfect square trinomials, identifying the pattern of the first and last terms being perfect squares and the middle term being twice the product of their square roots. This recognition allowed us to confidently apply the appropriate factoring formula. We then followed a step-by-step process, identifying 'a' and 'b', determining the correct sign, and applying the formula (a + b)² = a² + 2ab + b². This led us to the factored form (4x + 5)(4x + 5). To ensure accuracy, we verified our result by expanding the factored form, confirming that it matched the original trinomial.

The ability to factor perfect square trinomials efficiently is a valuable skill in algebra. It simplifies algebraic manipulations, aids in solving quadratic equations, and provides a foundation for more advanced mathematical concepts. By understanding the underlying pattern and following a systematic approach, we can confidently tackle these types of expressions. The process of factoring and verifying not only provides the correct answer but also deepens our understanding of the structure and relationships within algebraic expressions. This understanding is crucial for success in mathematics and related fields. Factoring is more than just a mechanical process; it's a powerful tool for understanding and manipulating the language of algebra.

Therefore, the correct answer is D. (4x + 5)(4x + 5)