Factoring 25z^2 - 95z + 60 A Step-by-Step Guide

In the realm of algebra, factoring quadratic expressions stands as a fundamental skill. It is a crucial technique used to simplify expressions, solve equations, and understand the behavior of quadratic functions. This comprehensive guide delves into the intricacies of factoring quadratic expressions, providing a step-by-step approach to mastering this essential algebraic skill. In this guide, we will address the given problem: Factor completely $25z^2 - 95z + 60$. If not factorable, write Prime.

Understanding Quadratic Expressions

Before we embark on the journey of factoring, it's imperative to grasp the essence of quadratic expressions. A quadratic expression is a polynomial of degree two, generally represented in the form $ax^2 + bx + c$, where 'a', 'b', and 'c' are constants, and 'x' is the variable. The coefficient 'a' cannot be zero, as that would reduce the expression to a linear one. The term $ax^2$ is the quadratic term, $bx$ is the linear term, and $c$ is the constant term. Understanding these components is the bedrock upon which our factoring skills will be built.

Factoring quadratic expressions involves decomposing the expression into a product of two binomials. These binomials, when multiplied together, yield the original quadratic expression. The ability to factor quadratic expressions hinges on recognizing patterns and applying the appropriate techniques. Several methods exist for factoring quadratic expressions, each with its own strengths and applications. We will explore the most common methods, including factoring out the greatest common factor (GCF), using the difference of squares pattern, and employing the AC method.

Step-by-Step Factoring Process

Let's embark on a step-by-step journey to factor the quadratic expression $25z^2 - 95z + 60$ completely. This process involves a series of systematic steps designed to break down the expression into its fundamental factors. By following these steps diligently, you'll be well-equipped to tackle a wide range of factoring problems.

1. Factoring out the Greatest Common Factor (GCF)

The first step in factoring any expression is to identify and factor out the greatest common factor (GCF). The GCF is the largest factor that divides all terms of the expression. In our given expression, $25z^2 - 95z + 60$, we need to find the largest number that divides 25, 95, and 60. By careful observation, we can determine that the GCF is 5. Factoring out the GCF, we get:

$5(5z^2 - 19z + 12)$

This step simplifies the expression and makes subsequent factoring steps easier to manage. Factoring out the GCF is a crucial first step in any factoring problem, as it often reduces the complexity of the expression and reveals underlying patterns.

2. Factoring the Remaining Quadratic Expression

After factoring out the GCF, we are left with the quadratic expression $5z^2 - 19z + 12$. Now, we need to factor this expression further. Since the coefficient of the $z^2$ term is not 1, we'll employ the AC method. The AC method involves finding two numbers that multiply to the product of the leading coefficient (A) and the constant term (C), and add up to the coefficient of the linear term (B). In this case, A = 5, B = -19, and C = 12. Therefore, we need to find two numbers that multiply to 5 * 12 = 60 and add up to -19.

By systematically considering the factors of 60, we can identify that the numbers -15 and -4 satisfy these conditions: (-15) * (-4) = 60 and (-15) + (-4) = -19. Now, we rewrite the middle term (-19z) using these two numbers:

$5z^2 - 15z - 4z + 12$

3. Factoring by Grouping

Next, we factor by grouping. We group the first two terms and the last two terms together:

$(5z^2 - 15z) + (-4z + 12)$

Now, we factor out the GCF from each group. From the first group, the GCF is 5z, and from the second group, the GCF is -4:

$5z(z - 3) - 4(z - 3)$

Notice that both terms now have a common binomial factor of (z - 3). We factor out this common factor:

$(z - 3)(5z - 4)$

4. Combining Factors

Finally, we combine the factors we've obtained. Remember that we initially factored out a GCF of 5. Now, we include this factor in our final factored expression:

$5(z - 3)(5z - 4)$

This is the completely factored form of the original quadratic expression.

Final Answer

Therefore, the completely factored form of $25z^2 - 95z + 60$ is $5(z - 3)(5z - 4)$.

Conclusion

Mastering the art of factoring quadratic expressions is a cornerstone of algebraic proficiency. By understanding the underlying principles, employing systematic techniques, and practicing diligently, you can confidently navigate the world of quadratic expressions and unlock their hidden structures. This comprehensive guide has provided you with the tools and knowledge to factor quadratic expressions effectively. Remember to always factor out the GCF first, use the AC method or other appropriate techniques, and factor by grouping when necessary. With consistent practice, you'll become adept at factoring quadratic expressions and excel in your algebraic endeavors.

This article has thoroughly explained the process of factoring quadratic expressions, using the specific example of $25z^2 - 95z + 60$. The step-by-step approach, from identifying the GCF to factoring by grouping, provides a clear and concise method for tackling such problems. By understanding the concepts and practicing the techniques outlined in this guide, you can confidently factor quadratic expressions and enhance your algebraic skills. Remember, factoring is not just a mathematical exercise; it's a fundamental tool that opens doors to deeper understanding and problem-solving in mathematics and beyond.