Factoring 36 - 4x² Completely A Step-by-Step Guide
Factoring quadratic expressions is a fundamental skill in algebra, and understanding how to do it thoroughly is crucial for solving equations and simplifying complex expressions. In this comprehensive guide, we will delve into the process of completely factoring the expression 36 - 4x². This involves identifying common factors, recognizing patterns like the difference of squares, and applying the appropriate techniques to arrive at the simplest factored form. Mastering these techniques will not only help you solve this particular problem but also equip you with the tools to tackle a wide range of factoring challenges.
Identifying the Greatest Common Factor (GCF)
The first step in factoring any algebraic expression is to identify and factor out the Greatest Common Factor (GCF). The GCF is the largest factor that divides evenly into all terms of the expression. In the given expression, 36 - 4x², we need to find the largest number and variable combination that can divide both 36 and 4x². Looking at the coefficients, 36 and 4, we can see that 4 is a common factor. Since the first term, 36, does not have a variable 'x', the GCF will not include any 'x' terms. Therefore, the GCF for 36 - 4x² is 4. Factoring out the GCF, we divide each term by 4:
36 / 4 = 9
-4x² / 4 = -x²
So, factoring out the GCF of 4 from the expression 36 - 4x² gives us:
4(9 - x²)
This simplifies the expression and makes it easier to recognize further factoring opportunities. Always remember to check for a GCF as the initial step in any factoring problem, as it simplifies the subsequent steps significantly. Recognizing the GCF not only makes the expression easier to handle but also ensures that the expression is completely factored, which is essential for many algebraic manipulations and problem-solving scenarios. For instance, when solving quadratic equations, completely factoring the expression allows us to easily identify the roots of the equation. Moreover, in calculus, simplifying expressions by factoring out the GCF can make differentiation and integration processes more manageable.
Recognizing the Difference of Squares Pattern
After factoring out the GCF, we are left with the expression 4(9 - x²). Now, we need to examine the expression inside the parentheses, (9 - x²), to see if it can be factored further. Here, we can recognize a specific pattern known as the difference of squares. The difference of squares pattern is a common and important factoring pattern in algebra. It states that the difference between two perfect squares can be factored into the product of the sum and difference of their square roots.
In mathematical terms, the difference of squares pattern is expressed as:
a² - b² = (a + b)(a - b)
To apply this pattern, we need to confirm that our expression fits the required form. In the expression (9 - x²), we can see that:
9 is a perfect square because it is 3² (3 squared).
x² is a perfect square because it is x² (x squared).
The operation between the two terms is subtraction, which signifies a 'difference'. Therefore, 9 - x² fits the difference of squares pattern. To factor it, we need to identify 'a' and 'b' in our expression. Comparing 9 - x² with a² - b², we can see that:
a² = 9, so a = √9 = 3
b² = x², so b = √x² = x
Now that we have identified 'a' and 'b', we can apply the difference of squares formula: a² - b² = (a + b)(a - b). Substituting a = 3 and b = x, we get:
9 - x² = (3 + x)(3 - x)
The recognition and application of the difference of squares pattern are crucial in factoring. This pattern frequently appears in algebraic problems, making its mastery essential for students. Being able to quickly identify this pattern can significantly simplify complex expressions and make problem-solving more efficient. Moreover, the difference of squares pattern is not just a tool for simplifying expressions; it also has applications in solving equations, particularly quadratic equations, and in simplifying rational expressions. Therefore, a solid understanding of the difference of squares pattern is a valuable asset in any algebraic context.
Applying the Difference of Squares Formula
Now that we have identified that the expression 9 - x² fits the difference of squares pattern, we can proceed to apply the formula. As discussed earlier, the difference of squares formula is:
a² - b² = (a + b)(a - b)
We've already established that in our expression 9 - x², a = 3 and b = x. Therefore, we can substitute these values into the formula:
9 - x² = (3 + x)(3 - x)
This step involves directly applying the formula by replacing 'a' and 'b' with their respective values. The expression is now factored into two binomials: (3 + x) and (3 - x). These binomials represent the sum and difference of the square roots of the original terms. By multiplying these two binomials, you would obtain the original expression 9 - x², confirming that we have factored it correctly. It is a good practice to mentally multiply the factors or use the FOIL (First, Outer, Inner, Last) method to verify that the factored form is equivalent to the original expression.
The application of the difference of squares formula is a straightforward process once the pattern is recognized and the values of 'a' and 'b' are correctly identified. This technique is not only applicable to simple expressions like 9 - x² but also to more complex expressions involving variables and coefficients. The key is to recognize the pattern and systematically apply the formula. Furthermore, understanding the logic behind the formula helps in remembering and applying it correctly. The difference of squares formula is derived from the distributive property of multiplication over addition and subtraction, which provides a deeper understanding of why this factoring pattern works. This method of factoring is a cornerstone in algebra and is used extensively in solving higher-level mathematical problems, including those in calculus and complex analysis.
Combining the GCF and Difference of Squares
We initially factored out the GCF of 4 from the expression 36 - 4x², which gave us 4(9 - x²). Then, we recognized and factored the difference of squares, 9 - x², as (3 + x)(3 - x). Now, to completely factor the original expression, we need to combine these two steps. This involves including the GCF that we factored out in the first step with the factored form we obtained from applying the difference of squares formula.
So, we have:
36 - 4x² = 4(9 - x²)
And we factored 9 - x² into (3 + x)(3 - x). Therefore, the completely factored form of 36 - 4x² is:
4(3 + x)(3 - x)
This is the final factored form of the expression. It is crucial to include the GCF in the final answer to ensure the expression is completely factored. Leaving out the GCF would mean that we have not factored the expression to its simplest form. The completely factored form provides a clear representation of the expression as a product of its factors, which is essential for solving equations, simplifying rational expressions, and other algebraic manipulations.
The process of combining the GCF and the difference of squares highlights the importance of a systematic approach to factoring. Starting with the GCF simplifies the expression and reveals other potential factoring patterns, such as the difference of squares. This step-by-step approach ensures that all factors are accounted for, leading to the completely factored form. Moreover, this process reinforces the understanding that factoring is the reverse operation of expansion or distribution. By combining the factors, we are essentially undoing the distribution process, resulting in the original expression. This concept is fundamental in algebra and is used extensively in various mathematical contexts.
Final Factored Form and Verification
Therefore, the completely factored form of the expression 36 - 4x² is:
4(3 + x)(3 - x)
To verify that our factored form is correct, we can expand the factored expression and check if it matches the original expression. Expanding the expression involves multiplying out the factors and simplifying the result. Let's start by multiplying the two binomial factors, (3 + x) and (3 - x):
(3 + x)(3 - x) = 3 * 3 + 3 * (-x) + x * 3 + x * (-x)
= 9 - 3x + 3x - x²
= 9 - x²
Now, we multiply this result by the GCF, which is 4:
4(9 - x²) = 4 * 9 - 4 * x²
= 36 - 4x²
This matches our original expression, 36 - 4x², which confirms that our factoring is correct. Verification is a critical step in the factoring process. It helps to ensure that the factored form is equivalent to the original expression, reducing the likelihood of errors. Expanding the factored form is one common method of verification, but other methods, such as substituting numerical values for the variable, can also be used. For example, we could substitute a value for x in both the original and factored expressions and see if the results are the same. If they are, it provides further evidence that our factoring is correct.
In conclusion, the complete factoring of the expression 36 - 4x² involves identifying and factoring out the GCF, recognizing and applying the difference of squares pattern, combining the results, and verifying the factored form. This process demonstrates the importance of a systematic approach to factoring and the significance of understanding fundamental factoring patterns. Mastering these techniques is essential for success in algebra and higher-level mathematics.
Alternative Representations of the Factored Form
While 4(3 + x)(3 - x) is the standard completely factored form of the expression 36 - 4x², it is worth noting that there are alternative ways to represent the same factored expression. These alternative representations are mathematically equivalent but may appear slightly different. Understanding these variations can be beneficial, especially when comparing answers or working with different algebraic manipulations.
One common alternative is to distribute the constant factor of 4 into one of the binomial factors. For example, we could distribute the 4 into the (3 - x) factor:
4(3 + x)(3 - x) = (3 + x) * 4(3 - x)
= (3 + x)(12 - 4x)
Alternatively, we could distribute the 4 into the (3 + x) factor:
4(3 + x)(3 - x) = 4(3 + x)(3 - x)
= (12 + 4x)(3 - x)
Both (3 + x)(12 - 4x) and (12 + 4x)(3 - x) are valid factored forms of the original expression. However, they are less simplified because the common factor of 4 is still present within one of the binomials. This is why the form 4(3 + x)(3 - x) is generally preferred, as it represents the most completely factored form.
Another variation involves rearranging the terms within the binomial factors or changing the signs. For example, we know that (a - b) is equivalent to -(b - a). Therefore, we can rewrite the (3 - x) factor as:
(3 - x) = -(x - 3)
Substituting this into our factored form, we get:
4(3 + x)(3 - x) = 4(3 + x)[-(x - 3)]
= -4(3 + x)(x - 3)
Similarly, we can rewrite the (3 + x) factor as (x + 3) because addition is commutative:
4(3 + x)(3 - x) = 4(x + 3)(3 - x)
These alternative representations highlight the flexibility in how factored expressions can be written. While some forms may be more conventional or simplified than others, understanding these variations can be helpful in recognizing equivalent expressions and manipulating them in different algebraic contexts. The key is to ensure that the factored form, regardless of its representation, is mathematically equivalent to the original expression. Verification through expansion or substitution remains a valuable tool in confirming the correctness of any factored form. This adaptability in representing factored expressions is a testament to the versatility and richness of algebraic manipulations.
