Factoring quadratic expressions is a fundamental skill in algebra. While y² + 16 might seem simple at first glance, understanding the process is crucial for tackling more complex problems. This guide provides clear, tangible steps to help you learn how to factorise y² + 16, and similar expressions, effectively.
Understanding Factorisation
Before diving into the specifics of y² + 16, let's establish a foundational understanding of factorisation. Factorisation is the process of breaking down an expression into smaller, simpler expressions that, when multiplied together, give you the original expression. Think of it like reverse multiplication. For example, factoring 6 would be 2 x 3.
The Difference of Two Squares
The key to factoring y² + 16 lies in recognising a specific pattern in algebra called the "difference of two squares". This pattern applies when you have an expression in the form: a² - b². This expression can always be factored as (a + b)(a - b).
Important Note: The difference of two squares only works when you are subtracting two perfect squares. This is where the challenge with y² + 16 comes in.
Why y² + 16 is Different
Notice that y² + 16 is a sum of two squares, not a difference. The difference of two squares formula, (a + b)(a - b), does not apply to sums of squares. y² + 16 cannot be factorised using real numbers.
Exploring Complex Numbers (Advanced)
While it's not typically covered at introductory algebra levels, it is possible to factor y² + 16 using complex numbers. Complex numbers involve the imaginary unit "i", where i² = -1. In this case, y² + 16 could be factored as (y + 4i)(y - 4i). However, for most basic algebra courses, the answer is simply that y² + 16 is irreducible (cannot be factored) over the real numbers.
Practicing Factorisation with Other Examples
To solidify your understanding of factorisation, let's try some examples where the difference of two squares does apply:
Example 1: Factor x² - 25
Here, a = x and b = 5. Applying the formula, we get (x + 5)(x - 5).
Example 2: Factor 4y² - 9
Here, a = 2y and b = 3. Applying the formula, we get (2y + 3)(2y - 3).
Key Takeaways:
- Difference of Two Squares: Remember the formula (a + b)(a - b) for factoring expressions of the form a² - b².
- Sum of Two Squares: Expressions of the form a² + b² generally cannot be factored using real numbers.
- Practice Makes Perfect: Work through numerous examples to develop your factoring skills.
By understanding the difference of two squares and its limitations, you can effectively approach and solve many quadratic factorisation problems. While y² + 16 presents a unique case that highlights the boundaries of real number factorisation, it ultimately strengthens your understanding of the process.
