Factoring Polynomials A Step By Step Guide To Solving 3x^2 + 30x + 75

Hey guys! Today, we're diving into a common algebra problem: factoring a polynomial expression. Specifically, we're going to tackle the expression $3x^2 + 30x + 75$. Factoring polynomials is a fundamental skill in algebra, and mastering it will help you solve equations, simplify expressions, and even tackle more advanced math topics later on. So, let's break it down step by step and make sure we understand exactly how to arrive at the correct answer.

Understanding the Basics of Factoring Polynomials

Before we jump into this specific problem, let's quickly recap what factoring actually means. Factoring a polynomial is like reverse multiplication. Think of it this way: when you multiply two expressions together, you get a new expression. Factoring is the process of taking that new expression and breaking it back down into its original components. It's like taking a cake and figuring out the recipe! For instance, if we multiply $(x + 2)$ and $(x + 3)$, we get $x^2 + 5x + 6$. Factoring, in this case, would involve starting with $x^2 + 5x + 6$ and breaking it down to $(x + 2)(x + 3)$. Got it? Great! Now, let’s see how this applies to our problem.

In our case, we have $3x^2 + 30x + 75$. The goal is to rewrite this as a product of simpler expressions. When you look at a polynomial like this, the first thing you should always do is check for a greatest common factor (GCF). The GCF is the largest number or expression that divides evenly into all the terms. Identifying and factoring out the GCF will simplify the polynomial, making it easier to factor the rest of the expression. So, let’s hunt for that GCF in $3x^2 + 30x + 75$. Looking at the coefficients 3, 30, and 75, we can see that they are all divisible by 3. Also, notice that the variable $x$ appears in the first two terms but not in the last term (75), so $x$ is not a common factor. Therefore, the GCF for this polynomial is 3.

Step-by-Step Factoring of $3x^2 + 30x + 75$

Okay, let’s get our hands dirty and start factoring. Remember, the expression we’re working with is $3x^2 + 30x + 75$.

1. Factor Out the Greatest Common Factor (GCF)

As we identified earlier, the GCF for this polynomial is 3. So, we factor out 3 from each term:

$3x^2 + 30x + 75 = 3(x^2 + 10x + 25)$

Notice how we've divided each term in the original polynomial by 3 and placed the 3 outside the parentheses. Now we have a simpler quadratic expression inside the parentheses: $x^2 + 10x + 25$. This is much easier to work with!

2. Factor the Quadratic Expression

Now, let’s focus on the expression inside the parentheses: $x^2 + 10x + 25$. This is a quadratic expression, which means it's in the form $ax^2 + bx + c$. In our case, $a = 1$, $b = 10$, and $c = 25$. To factor a quadratic expression, we’re looking for two numbers that multiply to give us $c$ (25) and add up to give us $b$ (10). Let's think about the factors of 25. We have:

• 1 and 25
• 5 and 5

Which pair adds up to 10? Bingo! It's 5 and 5. So, we can rewrite the quadratic expression as:

$x^2 + 10x + 25 = (x + 5)(x + 5)$

Notice that we have the same factor repeated twice, which means we can also write it as $(x + 5)^2$. This is a perfect square trinomial, which is a special type of quadratic expression that factors neatly into two identical binomials.

3. Combine the GCF and the Factored Quadratic

We’re almost there! Now we just need to put everything together. Remember, we factored out the GCF of 3 in the first step, and then we factored the quadratic expression. So, the complete factored form of the polynomial is:

$3x^2 + 30x + 75 = 3(x + 5)(x + 5)$

Or, equivalently:

$3x^2 + 30x + 75 = 3(x + 5)^2$

And there you have it! We've successfully factored the polynomial. Doesn't it feel good to solve a puzzle like that?

Analyzing the Answer Choices

Now that we've factored the polynomial, let's look at the answer choices provided and see which one matches our result.

• A. $(3x + 5)(x + 15)$
• B. $3(x - 5)(x + 5)$
• C. $3(x + 5)(x + 5)$
• D. $3(x - 5)(x - 5)$

Comparing our factored form, $3(x + 5)(x + 5)$, with the options, we can clearly see that option C is the correct answer.

Why Other Options are Incorrect

It's always helpful to understand why the other options are wrong. This helps solidify your understanding of the factoring process and prevents you from making similar mistakes in the future. Let’s quickly look at why options A, B, and D are incorrect:

• A. $(3x + 5)(x + 15)$: If we were to expand this expression, we would get $3x^2 + 50x + 75$, which is not the same as our original polynomial. The middle term is incorrect.
• B. $3(x - 5)(x + 5)$: This factors into $3(x^2 - 25)$, which simplifies to $3x^2 - 75$. This is not the same as our original polynomial. The middle term is missing, and the sign of the constant term is wrong.
• D. $3(x - 5)(x - 5)$: This factors into $3(x^2 - 10x + 25)$, which simplifies to $3x^2 - 30x + 75$. The middle term has the wrong sign.

By analyzing these incorrect options, we reinforce our understanding of how to correctly factor the polynomial.

Key Takeaways for Factoring Polynomials

Alright, guys, let's recap the key steps we took to factor the polynomial $3x^2 + 30x + 75$. Remembering these steps will make factoring other polynomials much easier:

1. Look for the Greatest Common Factor (GCF): Always start by identifying the GCF and factoring it out. This simplifies the expression and makes the subsequent factoring steps easier.
2. Factor the Quadratic Expression: After factoring out the GCF, focus on the remaining quadratic expression. Look for two numbers that multiply to give you the constant term and add up to give you the coefficient of the linear term.
3. Combine the GCF and the Factored Quadratic: Don’t forget to include the GCF in your final answer! It's a common mistake to factor the quadratic but forget about the GCF.
4. Check Your Answer: If you have time, you can always check your factored form by multiplying it out to see if you get back the original polynomial. This is a great way to catch any mistakes.

Practice Makes Perfect: Tips for Improving Your Factoring Skills

Factoring polynomials is a skill that gets better with practice. The more you do it, the more comfortable you'll become with recognizing patterns and applying the correct techniques. Here are a few tips to help you improve your factoring skills:

• Do lots of practice problems: The more polynomials you factor, the better you'll get. Start with simple examples and gradually work your way up to more complex ones. There are plenty of resources online and in textbooks where you can find practice problems.
• Recognize common factoring patterns: There are several common factoring patterns, such as the difference of squares ($a^2 - b^2 = (a - b)(a + b)$) and perfect square trinomials (like the one we saw in this problem). Learning to recognize these patterns will save you time and effort.
• Work with others: Factoring can be a challenging topic, so don’t be afraid to ask for help. Work with classmates, friends, or a tutor to discuss problems and share strategies. Explaining your thinking to someone else can also help solidify your understanding.
• Use online resources: There are many great websites and videos that can help you learn about factoring. Khan Academy, for example, has excellent videos and practice exercises on factoring polynomials.

Conclusion: Mastering Polynomial Factoring

So, there you have it! We’ve successfully factored the polynomial expression $3x^2 + 30x + 75$ and learned some valuable strategies for factoring polynomials in general. Remember, the key is to take it step by step, look for the GCF, factor the quadratic expression, and combine everything together. With practice and a solid understanding of the underlying concepts, you'll become a factoring master in no time! Keep practicing, stay curious, and you’ll conquer any polynomial that comes your way.

Factoring polynomials is not just an abstract math skill; it's a powerful tool that you'll use in many areas of mathematics and even in real-world applications. So, keep honing your skills, and you'll be well-equipped to tackle any algebraic challenge. You got this!