Factoring Expressions A Step-by-Step Guide

Hey guys! Today, we're diving into the world of factoring expressions, a fundamental skill in algebra that helps us simplify complex equations and make them easier to work with. We'll specifically tackle the expression ${\frac{30 z+120}{5 z+20}}$, breaking it down step-by-step so you can confidently factor similar problems on your own. Factoring is like reverse multiplication; instead of multiplying terms together, we're breaking them down into their constituent factors. This skill is super important not only for simplifying algebraic expressions but also for solving equations, graphing functions, and even in calculus. So, let's get started and demystify this essential algebraic technique!

Understanding the Basics of Factoring

Before we jump into our specific problem, let's quickly recap the basics of factoring. Factoring involves finding the common factors in an expression and pulling them out. Think of it like finding the greatest common divisor (GCD) but applied to algebraic terms. We aim to rewrite an expression as a product of its factors, making it simpler and revealing its underlying structure. When we talk about factoring, we are essentially looking for the largest expression (whether it's a number, a variable, or a combination) that divides evenly into all terms. The most common techniques include finding the greatest common factor (GCF), factoring by grouping, and using special product formulas such as the difference of squares or perfect square trinomials. The GCF is the largest factor that divides two or more numbers (or terms) without leaving a remainder. Factoring by grouping is useful when dealing with expressions containing four or more terms. Special product formulas help to quickly factor expressions that fit specific patterns. By mastering these techniques, you will be well-equipped to tackle a wide range of factoring problems, making algebra much more manageable and even, dare I say, fun! Factoring is not just a mathematical trick; it’s a way of understanding the structure of expressions and revealing hidden relationships. It lays the groundwork for more advanced topics and makes problem-solving a whole lot easier. So, let’s roll up our sleeves and get factoring!

Step-by-Step Factoring of the Numerator: ${30z + 120}$

Let's begin with the numerator, which is ${30z + 120}$. Our main goal here is to identify the greatest common factor (GCF) that both terms share. When you look at 30 and 120, what’s the biggest number that divides both of them? That’s right, it's 30! So, we can factor out 30 from both terms. Doing this involves dividing each term by the GCF and writing the expression as a product. To factor ${30z + 120}$, we'll start by finding the greatest common factor (GCF) of 30 and 120. The GCF is the largest number that divides both terms evenly. In this case, the GCF is 30. Now, we divide each term by the GCF: ${30z \div 30 = z}$ and ${120 \div 30 = 4}$. After identifying the GCF, we rewrite the expression as the product of the GCF and the remaining terms inside the parentheses. Factoring out 30 gives us: ${30(z + 4)}$. This means we rewrite ${30z + 120}$ as ${30 \times z + 30 \times 4}$, and then factor out the 30. When you factor out a common term, you’re essentially reversing the distributive property. Instead of multiplying the 30 across the terms inside the parentheses, we’re pulling it out. So, the factored form of the numerator ${30z + 120}$ is ${30(z + 4)}$. Make sure to always double-check your work by redistributing the factored term to ensure you arrive back at the original expression. This process of factoring out the GCF makes the expression simpler and helps us in further simplification or solving equations. Remember, factoring is a crucial skill in algebra, and mastering it will make handling more complex expressions and equations much easier.

So, we can rewrite the numerator as:

Numerator: ${30z + 120 = 30(z + 4)}$

Step-by-Step Factoring of the Denominator: ${5z + 20}$

Now, let's tackle the denominator, which is ${5z + 20}$. Just like with the numerator, we need to find the greatest common factor (GCF). Look at 5 and 20 – what's the largest number that divides both? You guessed it, it's 5! So, we'll factor out 5 from both terms. Factoring out the GCF simplifies the expression and helps in further steps, such as canceling common factors between the numerator and denominator. The denominator, ${5z + 20}$, also requires us to find the greatest common factor (GCF) to simplify it. The GCF is the largest number that divides both terms evenly. In this case, the GCF of 5 and 20 is 5. To factor ${5z + 20}$, we start by identifying the GCF, which is 5. Then, we divide each term by the GCF: ${5z \div 5 = z}$ and ${20 \div 5 = 4}$. Write the factored expression as the product of the GCF and the remaining terms inside the parentheses. Factoring out 5 gives us: ${5(z + 4)}$. When we divide each term by 5, we get ${z}$ from ${5z}$ and ${4}$ from ${20}$. So, we rewrite the expression as ${5 \times z + 5 \times 4}$ and factor out the 5. The factored form of the denominator ${5z + 20}$ is ${5(z + 4)}$. To verify, you can distribute the 5 back into the parentheses to ensure you get the original expression. This gives us ${5 \times z + 5 \times 4 = 5z + 20}$, which confirms our factoring is correct. By factoring out the GCF, we simplify the denominator, making it easier to work with in further calculations or simplifications.

So, we can rewrite the denominator as:

Denominator: ${5z + 20 = 5(z + 4)}$

Simplifying the Entire Expression

Alright, we've factored both the numerator and the denominator. Now comes the fun part – simplifying the entire expression! We started with ${\frac{30 z+120}{5 z+20}}$, and we've transformed it into ${\frac{30(z + 4)}{5(z + 4)}}$. Notice anything cool? Both the numerator and the denominator have a common factor of ${(z + 4)}$! This means we can cancel them out, just like simplifying fractions with common factors. We have successfully factored both the numerator and the denominator: Numerator: ${30z + 120 = 30(z + 4)}$ Denominator: ${5z + 20 = 5(z + 4)}$ Now, let's rewrite the original expression using these factored forms: ${\frac{30z + 120}{5z + 20} = \frac{30(z + 4)}{5(z + 4)}}$ Notice that both the numerator and denominator have a common factor of ${(z + 4)}$. This allows us to simplify the expression further by canceling out the common factor. We divide both the numerator and the denominator by ${(z + 4)}$: ${\frac{30(z + 4)}{5(z + 4)} = \frac{30}{5}}$ Now, we have a much simpler fraction to deal with. The final step is to reduce the fraction ${\frac{30}{5}}$. Both 30 and 5 are divisible by 5, so we divide both the numerator and the denominator by 5: ${\frac{30}{5} = \frac{30 \div 5}{5 \div 5} = \frac{6}{1}}$ So, the simplified fraction is ${\frac{6}{1}}$, which is simply 6. The simplified expression is 6. Factoring and canceling common factors makes the expression much easier to manage and understand. It’s like taking a complex puzzle and breaking it down into simple pieces.

Final Simplified Expression

After canceling out the ${(z + 4)}$ terms, we're left with ${\frac{30}{5}}$. What's 30 divided by 5? That's right, it's 6! So, the simplified expression is just 6. Isn't that neat? We started with a seemingly complex fraction and, through the magic of factoring, we've reduced it to a simple whole number. This skill of factoring and simplifying expressions is incredibly valuable in algebra and beyond. It allows us to tackle more complex problems with confidence and clarity. Remember, the key steps are to identify common factors, factor them out, and then cancel any common terms between the numerator and denominator. The final simplified expression is 6. To recap, we started with the expression ${\frac{30z + 120}{5z + 20}}$. We factored the numerator to ${30(z + 4)}$ and the denominator to ${5(z + 4)}$. We then rewrote the expression as ${\frac{30(z + 4)}{5(z + 4)}}$. By canceling the common factor of ${(z + 4)}$, we simplified the expression to ${\frac{30}{5}}$, which equals 6. This demonstrates the power of factoring in simplifying algebraic expressions and making them more manageable.

Therefore, the factored and simplified form of the expression ${\frac{30 z+120}{5 z+20}}$ is:

$$\frac{30 z+120}{5 z+20} = 6$$

Conclusion: Mastering Factoring Techniques

Great job, guys! We've successfully factored and simplified the expression ${\frac{30 z+120}{5 z+20}}$. We walked through each step, from finding the greatest common factors to canceling out terms, and arrived at the simplified answer of 6. Remember, the key to mastering factoring is practice, practice, practice! The more you work with different expressions, the more comfortable you'll become with identifying common factors and simplifying them. Factoring is a fundamental skill in algebra, enabling you to simplify complex expressions and solve equations more efficiently. It involves breaking down expressions into their simplest forms, which not only makes calculations easier but also reveals the underlying structure of mathematical problems. By mastering techniques such as finding the greatest common factor (GCF), factoring by grouping, and recognizing special product formulas, you can tackle a wide range of algebraic challenges. Each time you factor an expression, you're sharpening your problem-solving skills and building a deeper understanding of algebraic principles. Factoring is not just about manipulating symbols; it’s about understanding the relationships between numbers and variables. So, keep practicing, keep exploring, and keep challenging yourself with new problems. With consistent effort, you’ll become a factoring pro in no time! Keep practicing with various examples, and you'll soon find that factoring becomes second nature. This skill will be invaluable as you progress through your mathematical journey. Happy factoring!