Factor Completely: 25y^4 - 1 Explained Simply

by Alex Johnson 46 views

Let's break down how to factor the expression 25y^4 - 1 completely. Factoring is like reverse multiplication, and it's a crucial skill in algebra. We'll go step by step to make it easy to understand.

Understanding the Problem

Our mission is to factor the expression 25y^4 - 1. This means we want to rewrite it as a product of simpler expressions. The given expression is a difference of two squares, which is a common pattern that makes factoring easier. Identifying these patterns is key to quick and accurate factoring.

Recognizing the Difference of Squares

The expression 25y^4 - 1 fits the pattern a^2 - b^2, which factors into (a + b)(a - b). Here, a^2 is 25y^4 and b^2 is 1. Recognizing this pattern immediately sets us on the right path. The difference of squares is one of the most frequently used factoring patterns, so becoming familiar with it is highly beneficial.

Step-by-Step Factoring

Now, let's factor 25y^4 - 1 step by step:

  1. Identify a and b:

    • Since a^2 = 25y^4, then a = 5y^2.
    • Since b^2 = 1, then b = 1.

  2. Apply the Difference of Squares Formula:

    • Using a^2 - b^2 = (a + b)(a - b), we get: 25y^4 - 1 = (5y^2 + 1)(5y^2 - 1)

  3. Check for Further Factoring:

    • Now, we examine the factors (5y^2 + 1) and (5y^2 - 1) to see if they can be factored further.

    • The term (5y^2 + 1) is a sum of squares and generally cannot be factored further using real numbers. Sums of squares do not factor in the same straightforward way that differences of squares do.

    • However, (5y^2 - 1) is again a difference of squares, but this time it involves a bit more attention since 5 is not a perfect square. We can rewrite it as (√5y)^2 - 1^2.

  4. Factor (5y^2 - 1):

    • Applying the difference of squares formula again: 5y^2 - 1 = (√5y + 1)(√5y - 1)

  5. Combine All Factors:

    • So, the completely factored expression is: 25y^4 - 1 = (5y^2 + 1)(√5y + 1)(√5y - 1)

Detailed Explanation

To ensure clarity, let’s delve deeper into each step. Understanding the 'why' behind each manipulation is as important as the steps themselves.

Identifying 'a' and 'b'

In the difference of squares pattern a^2 - b^2, correctly identifying 'a' and 'b' is crucial. For 25y^4 - 1, we recognize that 25y^4 is a perfect square because 25 is 5^2 and y^4 is (y2)2. Therefore, a = 5y^2. Similarly, 1 is a perfect square (1^2 = 1), so b = 1. Always double-check that your 'a' and 'b' values, when squared, match the original terms.

Applying the Difference of Squares Formula

Once 'a' and 'b' are identified, applying the formula a^2 - b^2 = (a + b)(a - b) is straightforward. Substituting a = 5y^2 and b = 1, we get 25y^4 - 1 = (5y^2 + 1)(5y^2 - 1). This step transforms the original expression into a product of two binomials.

Checking for Further Factoring

After the first application of the difference of squares, it’s essential to check whether either of the resulting factors can be factored further. The term (5y^2 + 1) is a sum of squares, and it does not factor nicely using real numbers. However, (5y^2 - 1) is another difference of squares, albeit less obvious because 5 is not a perfect square. Recognizing these nested patterns is a key aspect of mastering factoring.

Factoring (5y^2 - 1) with Square Roots

To factor (5y^2 - 1), we recognize it as (√5y)^2 - 1^2. Applying the difference of squares formula again, we get 5y^2 - 1 = (√5y + 1)(√5y - 1). This might seem unusual because of the square roots, but it is mathematically valid and completes the factoring process.

Combining All Factors for the Final Result

Finally, we combine all the factors to get the completely factored expression: 25y^4 - 1 = (5y^2 + 1)(√5y + 1)(√5y - 1). This is the most simplified form of the original expression, written as a product of its factors.

Common Mistakes to Avoid

  • Incorrectly Identifying a and b: Ensure you take the square root correctly. For example, if a^2 = 25y^4, then a = 5y^2, not 25y^2 or 5y^4.
  • Forgetting to Check for Further Factoring: Always examine the resulting factors to see if they can be factored further. This is especially important when dealing with expressions that have multiple layers of factoring.
  • Incorrectly Applying the Difference of Squares Formula: Double-check that you are adding and subtracting the correct terms in the (a + b)(a - b) format.
  • Mistaking Sum of Squares for Difference of Squares: A sum of squares (a^2 + b^2) generally does not factor using real numbers, unlike a difference of squares (a^2 - b^2).

Practice Problems

To solidify your understanding, try factoring these expressions:

  1. 16x^4 - 1
  2. 81a^4 - b^4
  3. 4p^4 - 9

Conclusion

Factoring the expression 25y^4 - 1 involves recognizing and applying the difference of squares pattern twice. The completely factored form is (5y^2 + 1)(√5y + 1)(√5y - 1). By understanding the underlying principles and avoiding common mistakes, you can confidently tackle similar factoring problems. Remember to always look for patterns and check for further factoring to arrive at the simplest form.

For more information on factoring and algebraic techniques, you can visit Khan Academy's Algebra Section.