Factoring Binomials A Comprehensive Guide To $25p^3q - 36pq^3$

Factoring binomials is a fundamental skill in algebra, allowing us to simplify expressions, solve equations, and gain deeper insights into mathematical relationships. In this comprehensive guide, we will delve into the process of factoring the binomial $25p^3q - 36pq^3$ completely. This exploration will not only provide a step-by-step solution but also equip you with the knowledge to tackle similar problems with confidence.

Understanding the Basics of Factoring

Before we dive into the specifics of our binomial, it's crucial to grasp the fundamental concepts of factoring. Factoring is the process of breaking down a mathematical expression into its constituent factors, which, when multiplied together, yield the original expression. In simpler terms, it's the reverse of expansion. For instance, factoring the number 12 gives us $2 × 2 × 3$, as these numbers multiply to 12.

In algebra, we often deal with polynomials, which are expressions consisting of variables and coefficients. Factoring polynomials involves identifying common factors among the terms and expressing the polynomial as a product of simpler expressions. This process is essential for solving polynomial equations, simplifying algebraic fractions, and performing various other algebraic manipulations.

The greatest common factor (GCF) plays a pivotal role in factoring. The GCF is the largest factor that divides all terms of the polynomial. Identifying and extracting the GCF is the first step in many factoring problems, as it simplifies the expression and makes further factorization easier.

Step-by-Step Factoring of $25p^3q - 36pq^3$

Now, let's apply these concepts to factor the binomial $25p^3q - 36pq^3$ completely. We'll break down the process into manageable steps to ensure clarity and understanding.

1. Identify the Greatest Common Factor (GCF)

The first step in factoring any polynomial is to identify the GCF of its terms. In the binomial $25p^3q - 36pq^3$, we need to find the factors common to both terms.

• Numerical Coefficients: The numerical coefficients are 25 and 36. The greatest common factor of 25 and 36 is 1, as they share no common factors other than 1.
• Variable Factors: The variable factors are $p^3q$ and $pq^3$. To find the GCF of the variable factors, we take the lowest power of each common variable. Both terms have p and q. The lowest power of p is $p^1$ (or simply p), and the lowest power of q is $q^1$ (or simply q). Therefore, the GCF of the variable factors is pq.

Combining the GCF of the numerical coefficients and the variable factors, we find that the GCF of the entire binomial is $1 * pq$, which simplifies to pq.

2. Factor out the GCF

Now that we've identified the GCF as pq, we can factor it out from the binomial. This involves dividing each term of the binomial by the GCF and writing the result in parentheses.

25p^3q - 36pq^3 = pq( rac{25p^3q}{pq} - rac{36pq^3}{pq})

Simplifying the fractions inside the parentheses, we get:

$25p^3q - 36pq^3 = pq(25p^2 - 36q^2)$

3. Recognize the Difference of Squares

After factoring out the GCF, we are left with the expression $(25p^2 - 36q^2)$ inside the parentheses. This expression is a difference of squares, a special type of binomial that can be factored further.

A difference of squares has the form $a^2 - b^2$, where a and b are algebraic expressions. It can be factored as $(a + b)(a - b)$.

In our case, $25p^2$ can be written as $(5p)^2$ and $36q^2$ can be written as $(6q)^2$. Thus, we have a difference of squares where $a = 5p$ and $b = 6q$.

4. Factor the Difference of Squares

Applying the difference of squares factorization formula, we can factor $(25p^2 - 36q^2)$ as follows:

$25p^2 - 36q^2 = (5p + 6q)(5p - 6q)$

5. Write the Complete Factorization

Finally, we combine the GCF we factored out earlier with the factorization of the difference of squares to obtain the complete factorization of the binomial.

$25p^3q - 36pq^3 = pq(25p^2 - 36q^2) = pq(5p + 6q)(5p - 6q)$

Therefore, the completely factored form of the binomial $25p^3q - 36pq^3$ is $pq(5p + 6q)(5p - 6q)$.

Importance of Factoring

Factoring is not merely an algebraic exercise; it's a powerful tool with numerous applications in mathematics and beyond. Understanding how to factor expressions allows us to:

• Solve Equations: Factoring is crucial for solving polynomial equations. By setting a factored polynomial equal to zero, we can use the zero-product property to find the roots or solutions of the equation.
• Simplify Expressions: Factoring helps in simplifying complex algebraic expressions, making them easier to work with and understand.
• Solve Real-World Problems: Many real-world problems can be modeled using polynomial equations. Factoring enables us to solve these equations and find meaningful solutions.
• Understand Mathematical Relationships: Factoring reveals the underlying structure of mathematical expressions, providing insights into their properties and relationships.

Common Factoring Techniques

While we focused on factoring out the GCF and the difference of squares in this example, several other factoring techniques are essential for mastering algebra. Some common techniques include:

• Factoring by Grouping: This technique is used for polynomials with four or more terms. It involves grouping terms together and factoring out common factors from each group.
• Factoring Trinomials: Trinomials, which have three terms, can often be factored into the product of two binomials. Techniques like the AC method or trial and error are used for factoring trinomials.
• Sum and Difference of Cubes: Expressions in the form of $a^3 + b^3$ or $a^3 - b^3$ can be factored using specific formulas.

Conclusion

In conclusion, factoring the binomial $25p^3q - 36pq^3$ completely involves identifying the GCF, recognizing the difference of squares pattern, and applying the appropriate factoring techniques. The completely factored form is $pq(5p + 6q)(5p - 6q)$. Mastering factoring skills is crucial for success in algebra and higher-level mathematics. By understanding the underlying concepts and practicing various factoring techniques, you can confidently tackle a wide range of factoring problems and unlock the power of algebraic manipulation.