Factor $48y^2 - 27$ Completely

When we talk about factoring polynomials, especially expressions like $48y^2 - 27$, we're essentially looking to break them down into their simplest multiplicative components. This process is fundamental in algebra and can make complex equations much more manageable. The goal is to find two or more expressions that, when multiplied together, give us the original expression. Think of it like finding the prime factors of a number, but for algebraic terms. For $48y^2 - 27$, the journey to complete factorization involves a few key steps, and understanding each one is crucial. We'll be looking for common factors, differences of squares, and potentially other patterns that reveal themselves as we simplify. The beauty of complete factorization is that it ensures we've gone as far as possible in breaking down the expression, which is essential for solving equations, simplifying fractions, and many other mathematical operations. So, let's dive into how we can systematically approach factoring this specific expression.

Identifying the Greatest Common Factor (GCF)

The very first step in factoring any polynomial, including $48y^2 - 27$, is to identify and extract the Greatest Common Factor (GCF). The GCF is the largest term that divides into every term in the expression without leaving a remainder. For $48y^2$ and $27$, we need to find the largest number that goes into both 48 and 27, and also consider any common variables. Looking at the coefficients, 48 and 27, we can see that they are both divisible by 3. (48 / 3 = 16, and 27 / 3 = 9). There isn't a larger integer that divides both 48 and 27. Now, let's consider the variables. The term $48y^2$ has $y^2$, while the term $27$ has no variable part. Since there's no common variable across both terms, the GCF is simply the numerical value, which is 3. Extracting the GCF involves dividing each term in the original expression by the GCF and then writing the expression as the GCF multiplied by the result of this division. So, for $48y^2 - 27$, we would have: $3( (48y^2)/3 - (27)/3 )$. This simplifies to $3(16y^2 - 9)$. This initial step of factoring out the GCF is absolutely critical, as it not only simplifies the expression but also often reveals underlying structures that can be factored further in subsequent steps. If we were to miss this GCF, any further factoring attempts might not lead to the complete factorization, which is our ultimate goal. It's a foundational move that sets the stage for uncovering more complex factoring patterns.

Recognizing the Difference of Squares

After factoring out the GCF, we are left with the expression $3(16y^2 - 9)$. Now, we need to examine the expression inside the parentheses: $16y^2 - 9$. This part of the expression exhibits a very common and useful algebraic pattern: the difference of squares. The difference of squares pattern applies to any expression in the form of $a^2 - b^2$, which can always be factored into $(a - b)(a + b)$. To see if $16y^2 - 9$ fits this pattern, we need to determine if both terms are perfect squares. The first term, $16y^2$, is a perfect square because $16$ is $4^2$ and $y^2$ is $y^2$. Therefore, $16y^2$ can be written as $(4y)^2$. The second term, $9$, is also a perfect square, as $9 = 3^2$. Since we have a perfect square ($ (4y)^2 $) minus another perfect square ($3^2$), we can apply the difference of squares formula. Here, $a = 4y$ and $b = 3$. So, $16y^2 - 9$ can be factored as $(4y - 3)(4y + 3)$. It's important to be comfortable identifying these perfect squares and applying the formula correctly. Missing this pattern means we haven't achieved complete factorization. The structure of the problem often guides us; the minus sign between two terms that are perfect squares is a strong indicator. Once identified, the factorization is straightforward and leads us to the final factored form of the original expression.

Final Complete Factorization

Now, let's bring it all together. We started with the expression $48y^2 - 27$. Our first step was to factor out the GCF, which we found to be 3, leaving us with $3(16y^2 - 9)$. In the second step, we recognized that the expression inside the parentheses, $16y^2 - 9$, is a difference of squares. We identified $a^2 = 16y^2$ (so $a = 4y$) and $b^2 = 9$ (so $b = 3$). Applying the difference of squares formula ($a^2 - b^2 = (a - b)(a + b)$), we factored $16y^2 - 9$ into $(4y - 3)(4y + 3)$. Now, we substitute this back into our expression with the GCF. The complete factorization of $48y^2 - 27$ is therefore $3(4y - 3)(4y + 3)$. To verify this, we can multiply the factors back together. First, multiply the binomials: $(4y - 3)(4y + 3) = (4y)(4y) + (4y)(3) - (3)(4y) - (3)(3) = 16y^2 + 12y - 12y - 9 = 16y^2 - 9$. Then, multiply this result by the GCF, 3: $3(16y^2 - 9) = 3(16y^2) - 3(9) = 48y^2 - 27$. This matches our original expression, confirming that our factorization is indeed complete and correct. Mastering these steps – finding the GCF and recognizing patterns like the difference of squares – is key to efficiently factoring various algebraic expressions.

Conclusion

Factoring polynomials completely, as demonstrated with $48y^2 - 27$, is a crucial skill in mathematics that involves breaking down expressions into their most basic multiplicative parts. We successfully factored $48y^2 - 27$ by first identifying and extracting the Greatest Common Factor (GCF), which was 3. This simplified the expression to $3(16y^2 - 9)$. Subsequently, we recognized that the term within the parentheses, $16y^2 - 9$, is a difference of squares. Applying the formula $a^2 - b^2 = (a - b)(a + b)$, where $a = 4y$ and $b = 3$, we factored $16y^2 - 9$ into $(4y - 3)(4y + 3)$. Combining these steps, the complete factorization of $48y^2 - 27$ is $3(4y - 3)(4y + 3)$. This systematic approach ensures that we have reduced the expression to its simplest form, which is invaluable for solving equations, simplifying algebraic fractions, and understanding more advanced mathematical concepts. Consistent practice with identifying GCFs and recognizing common factoring patterns like the difference of squares will enhance your algebraic proficiency. For further exploration into factoring techniques and algebraic manipulation, you can visit Khan Academy's algebra section or delve into the resources provided by the National Council of Teachers of Mathematics (NCTM).