Factoring (p^2-4)(9-q^2)+24pq A Comprehensive Guide

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Introduction

In this detailed discussion, we will explore the process of factoring the algebraic expression $(p^2-4)(9-q^2)+24pq$. This problem falls under the category of mathematics, specifically algebra, and involves techniques such as expanding, rearranging terms, and recognizing patterns to simplify and factor the given expression. Factoring is a fundamental skill in algebra, often used to solve equations, simplify expressions, and analyze mathematical relationships. Our goal is to break down the expression into simpler components, which can provide deeper insights into its structure and behavior. This comprehensive guide will walk you through each step, ensuring a clear understanding of the methods and principles applied.

Expanding the Expression

The first step in factoring the expression $(p^2-4)(9-q^2)+24pq$ is to expand the product of the two binomials. This involves applying the distributive property (also known as the FOIL method for binomials), which states that each term in the first binomial must be multiplied by each term in the second binomial. This process helps to eliminate the parentheses and combine like terms, making the expression easier to manipulate. Let's dive into the expansion:

Expanding $(p^2-4)(9-q^2)$:

$(p^2-4)(9-q^2) = p^2(9) + p^2(-q^2) - 4(9) - 4(-q^2)$

Simplifying each term:

$p^2(9) = 9p^2$ $p^2(-q^2) = -p^2q^2$ $-4(9) = -36$ $-4(-q^2) = 4q^2$

Combining these terms, we get:

$(p^2-4)(9-q^2) = 9p^2 - p^2q^2 - 36 + 4q^2$

Now, we add the remaining term from the original expression, which is $24pq$:

$9p^2 - p^2q^2 - 36 + 4q^2 + 24pq$

This expanded form provides a clearer picture of the terms we need to work with. The next step involves rearranging the terms to identify potential patterns or groupings that can lead to factorization. This process is crucial for simplifying complex algebraic expressions and requires a keen eye for recognizing structures that can be factored.

Rearranging Terms and Recognizing Patterns

After expanding the expression, the next critical step is to rearrange the terms in a manner that reveals underlying patterns and facilitates factorization. In the expanded form $9p^2 - p^2q^2 - 36 + 4q^2 + 24pq$, a strategic rearrangement can help us identify potential groupings or perfect square trinomials. By carefully positioning terms, we can transform the expression into a more manageable form for factoring.

Let's rearrange the terms to group similar terms together and highlight potential patterns:

$-p^2q^2 + 9p^2 + 24pq + 4q^2 - 36$

This rearrangement brings terms with similar variables and degrees closer together. Now, let's consider grouping the terms to form a recognizable pattern. We can rewrite the expression as:

$(-p^2q^2 + 24pq) + (9p^2 + 4q^2 - 36)$

However, this grouping doesn't immediately reveal a clear factorization. A more insightful rearrangement involves considering a quadratic-like structure. Let's rearrange the terms once more:

$9p^2 + 24pq + 4q^2 - p^2q^2 - 36$

Notice that the first three terms, $9p^2 + 24pq + 4q^2$, resemble a perfect square trinomial. A perfect square trinomial is of the form $(ax + by)^2$ or $(ax - by)^2$. In this case, we have:

$(3p)^2 + 2(3p)(2q) + (2q)^2$

This is indeed a perfect square trinomial, which can be factored as $(3p + 2q)^2$. So, we can rewrite the expression as:

$(3p + 2q)^2 - p^2q^2 - 36$

This form is significantly more structured and brings us closer to a factorable expression. Recognizing these patterns is crucial in algebra, as it allows us to apply standard factoring techniques effectively. The next step will involve further manipulation to factor the expression completely.

Factoring as a Difference of Squares

Having rearranged and recognized the perfect square trinomial in the expression $(3p + 2q)^2 - p^2q^2 - 36$, the next strategic step is to identify if the expression can be further factored using the difference of squares. The difference of squares is a common algebraic identity, $a^2 - b^2 = (a + b)(a - b)$, which can greatly simplify expressions. To apply this, we need to manipulate the current form to fit the $a^2 - b^2$ pattern.

First, let's rewrite the expression to group the last two terms:

$(3p + 2q)^2 - (p^2q^2 + 36)$

However, this form does not directly fit the difference of squares pattern. To achieve this, we need to consider an alternative rearrangement. Let's revisit the expression:

$(3p + 2q)^2 - p^2q^2 - 36$

We can rewrite 36 as $6^2$. Now, the expression looks like:

$(3p + 2q)^2 - p^2q^2 - 6^2$

This still doesn't fit the difference of squares pattern directly. Instead, let's try grouping the terms differently. We can rewrite the expression as:

$(3p + 2q)^2 - (pq)^2 - 6^2$

Now, this isn't immediately factorable as a simple difference of squares. However, let’s try another approach. Consider the original expression before this step:

$9p^2 + 24pq + 4q^2 - p^2q^2 - 36$

We can rearrange this as:

$9p^2 + 24pq + 4q^2 - (p^2q^2 + 36)$

And from our previous perfect square trinomial identification:

$(3p + 2q)^2 - (p^2q^2 + 36)$

This still doesn’t fit. Let’s look at another manipulation:

Rearrange the terms as:

$9p^2 - 36 + 24pq + 4q^2 - p^2q^2$

Group the terms:

$(9p^2 - 36) + (24pq + 4q^2 - p^2q^2)$

Factor out 9 from the first group:

$9(p^2 - 4) + (24pq + 4q^2 - p^2q^2)$

This also does not lead to an immediate factorization using the difference of squares. Let’s go back and reconsider our perfect square trinomial approach:

$(3p + 2q)^2 - p^2q^2 - 36$

We need to recognize a clever manipulation here. Let's treat the first two terms as a difference of squares:

$[(3p + 2q)^2 - (pq)^2] - 36$

Now we have a difference of squares, where $a^2 = (3p + 2q)^2$ and $b^2 = (pq)^2$. Applying the difference of squares formula, we get:

$[(3p + 2q) + pq][(3p + 2q) - pq] - 36$

Expanding the terms inside the brackets:

$(3p + 2q + pq)(3p + 2q - pq) - 36$

Now, we have a more complex expression. Let's evaluate whether this can be simplified further.

Final Factoring Steps

After applying the difference of squares, we arrived at the expression $(3p + 2q + pq)(3p + 2q - pq) - 36$. The final step in factoring involves determining if we can further simplify or factor this expression. At this point, it's crucial to carefully analyze the terms and look for any remaining patterns or common factors.

Let’s examine the expression closely:

$(3p + 2q + pq)(3p + 2q - pq) - 36$

We have two trinomials multiplied together, and then we are subtracting 36. This does not immediately reveal any obvious factoring patterns. Expanding the product of the two trinomials could potentially help, but it might also lead to a more complex expression. Let's try expanding the product:

$(3p + 2q + pq)(3p + 2q - pq) = (3p + 2q)^2 - (pq)^2$

We’ve already seen this pattern before. Let's expand $(3p + 2q)^2$:

$(3p + 2q)^2 = (3p)^2 + 2(3p)(2q) + (2q)^2 = 9p^2 + 12pq + 4q^2$

So, the expanded product of the two trinomials is:

$9p^2 + 12pq + 4q^2 - p^2q^2$

Now, we subtract 36 from this result:

$9p^2 + 12pq + 4q^2 - p^2q^2 - 36$

Comparing this with our earlier expanded form of the original expression, which was:

$9p^2 - p^2q^2 - 36 + 4q^2 + 24pq$

We see that there might have been a mistake in our expansion or factoring process, as the $12pq$ term here does not match the $24pq$ term in the original expanded expression. Let’s go back and verify each step to find the error.

Revisiting the Expansion and Rearrangement:

The original expression was:

$(p^2 - 4)(9 - q^2) + 24pq$

Expanding:

$9p^2 - p^2q^2 - 36 + 4q^2 + 24pq$

Rearranging:

$9p^2 + 24pq + 4q^2 - p^2q^2 - 36$

Perfect square trinomial:

$(3p + 2q)^2 - p^2q^2 - 36$

Difference of squares (incorrect application):

$[(3p + 2q + pq)(3p + 2q - pq)] - 36$

Expanding (where the error likely occurred):

Let’s expand $(3p + 2q + pq)(3p + 2q - pq)$ correctly:

Using the difference of squares, $(a + b)(a - b) = a^2 - b^2$, where $a = (3p + 2q)$ and $b = pq$:

$(3p + 2q + pq)(3p + 2q - pq) = (3p + 2q)^2 - (pq)^2$

$(3p + 2q)^2 = 9p^2 + 12pq + 4q^2$

$(pq)^2 = p^2q^2$

So, $(3p + 2q + pq)(3p + 2q - pq) = 9p^2 + 12pq + 4q^2 - p^2q^2$

Now, subtracting 36:

$9p^2 + 12pq + 4q^2 - p^2q^2 - 36$

Comparing this with the original expanded form:

$9p^2 - p^2q^2 - 36 + 4q^2 + 24pq$

We notice a discrepancy: the $12pq$ term in our expansion does not match the $24pq$ term in the original expression. This indicates an error in our intermediate steps. Let’s backtrack to the perfect square trinomial and reconsider our approach.

Returning to the Perfect Square Trinomial:

We have $(3p + 2q)^2 - p^2q^2 - 36$. A crucial insight is to rearrange the original expanded expression slightly differently:

$9p^2 + 24pq + 4q^2 - 36 - p^2q^2$

We identified $(3p + 2q)^2 = 9p^2 + 12pq + 4q^2$, but the original expression has $24pq$, not $12pq$. This suggests we cannot directly form a perfect square trinomial with all the terms involving $p$ and $q$.

Another Approach:

Let’s try to group terms in a different way. We can rewrite the expression as:

$9p^2 + 24pq + 4q^2 - p^2q^2 - 36$

Consider factoring by grouping. This method involves identifying common factors among groups of terms. However, in this expression, there isn't an obvious grouping that leads to a direct factorization. Alternatively, we might look for a substitution or a different algebraic identity that can simplify the expression. But let's rethink from the beginning.

Original Expression:

$(p^2 - 4)(9 - q^2) + 24pq$

Let's factor each term inside the parentheses if possible.

$(p^2 - 4)$ is a difference of squares, so it factors into $(p - 2)(p + 2)$. $(9 - q^2)$ is also a difference of squares, so it factors into $(3 - q)(3 + q)$.

Substituting these back into the original expression:

$(p - 2)(p + 2)(3 - q)(3 + q) + 24pq$

This factored form of the first part of the expression is a good starting point. Now, we need to integrate the $24pq$ term and see if we can find a way to combine it with the factored terms. Expanding this product completely could give us a better insight.

Expanding the Product:

$(p - 2)(p + 2)(3 - q)(3 + q) = (p^2 - 4)(9 - q^2) = 9p^2 - p^2q^2 - 36 + 4q^2$

Adding the $24pq$ term:

$9p^2 - p^2q^2 - 36 + 4q^2 + 24pq$

We're back to our initial expanded form. This confirms that our expansion was correct, and the challenge lies in finding the right factorization method.

Let's try a Different Grouping Approach:

Rearrange the terms to create a quadratic-like form:

$-p^2q^2 + 24pq + 9p^2 + 4q^2 - 36$

If we consider this as a quadratic in $pq$, it doesn't immediately suggest a factorization. However, let’s try to complete the square with the terms that include $pq$. This approach can sometimes reveal hidden structures in algebraic expressions. The given expression remains a challenge, indicating there could be an error in the initial problem statement or that the factorization requires a less obvious algebraic manipulation. After several attempts using standard factoring techniques, we arrive at the conclusion that a straightforward factorization might not be possible, and advanced techniques or a different approach may be required to further simplify the expression.

Conclusion

In conclusion, the process of factoring the expression $(p^2-4)(9-q^2)+24pq$ involves a series of algebraic manipulations, including expansion, rearrangement, and recognition of patterns such as the difference of squares and perfect square trinomials. Despite these efforts, a straightforward factorization is not immediately apparent. The complexity of the expression suggests that there may not be a simple factored form, or that advanced techniques beyond basic algebraic manipulation are required. This exercise highlights the importance of persistence and adaptability in problem-solving, as well as the understanding that not all algebraic expressions can be factored neatly using elementary methods. The detailed walkthrough of the steps demonstrates the systematic approach needed to tackle complex mathematical problems, emphasizing the value of meticulous verification and exploration of alternative strategies.