Factoring Polynomials $6x^4 - 30x^3 - 84x^2$ A Step-by-Step Guide

Hey guys! Let's dive into the world of factoring polynomials. It might seem daunting at first, but trust me, with a systematic approach, it becomes quite manageable. Today, we're going to tackle the polynomial $6x^4 - 30x^3 - 84x^2$ and break it down into its completely factored form. So, grab your thinking caps, and let's get started!

1. Understanding Factoring and Why It Matters

Before we jump into the problem, let's quickly recap what factoring is and why it's so important in mathematics. In simple terms, factoring a polynomial means expressing it as a product of simpler polynomials or expressions. Think of it like breaking down a number into its prime factors – instead of numbers, we're dealing with algebraic expressions.

So, why bother factoring? Well, factoring is a fundamental skill in algebra and calculus. It's used extensively in solving equations, simplifying expressions, finding roots of polynomials, and even in more advanced topics like calculus and differential equations. Mastering factoring is like unlocking a superpower in your mathematical toolkit!

Factoring helps us simplify complex expressions, making them easier to work with. For instance, imagine trying to solve an equation like $6x^4 - 30x^3 - 84x^2 = 0$ directly. It looks pretty intimidating, right? But if we can factor the left-hand side, the equation becomes much simpler to solve. Factoring can also reveal key information about the polynomial, such as its roots (the values of x that make the polynomial equal to zero) and its behavior.

In various real-world applications, polynomials often represent physical quantities or relationships. Factoring these polynomials can provide insights into the underlying system. For example, in physics, polynomials might describe the trajectory of a projectile, and factoring can help determine the time it takes to reach a certain point. Similarly, in engineering, polynomials can model the behavior of electrical circuits, and factoring can aid in analyzing the circuit's stability.

2. Identifying the Greatest Common Factor (GCF)

The first step in factoring any polynomial is to look for the Greatest Common Factor (GCF). The GCF is the largest factor that divides evenly into all the terms of the polynomial. It's like finding the biggest piece you can pull out from every part of the expression.

In our polynomial, $6x^4 - 30x^3 - 84x^2$, we need to identify the GCF of the coefficients (6, -30, and -84) and the variables ($x^4$, $x^3$, and $x^2$). Let's start with the coefficients.

The factors of 6 are 1, 2, 3, and 6. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84. Looking at these lists, we can see that the greatest common factor of 6, 30, and 84 is 6. So, 6 is part of our GCF.

Now, let's consider the variables. We have $x^4$, $x^3$, and $x^2$. The GCF of variables with exponents is the variable raised to the lowest exponent present in the terms. In this case, the lowest exponent is 2 (from $x^2$). So, $x^2$ is also part of our GCF.

Combining the GCF of the coefficients and the variables, we find that the GCF of the entire polynomial $6x^4 - 30x^3 - 84x^2$ is $6x^2$. This is the key that unlocks our factoring journey!

3. Factoring out the GCF

Now that we've identified the GCF as $6x^2$, the next step is to factor it out from the polynomial. This means dividing each term of the polynomial by $6x^2$ and writing the result in parentheses.

Let's do it step by step:

• Divide the first term, $6x^4$, by $6x^2$: $(6x^4) / (6x^2) = x^2$
• Divide the second term, $-30x^3$, by $6x^2$: $(-30x^3) / (6x^2) = -5x$
• Divide the third term, $-84x^2$, by $6x^2$: $(-84x^2) / (6x^2) = -14$

Now, we can rewrite the original polynomial as the GCF multiplied by the expression in parentheses:

$6x^4 - 30x^3 - 84x^2 = 6x^2(x^2 - 5x - 14)$

We've successfully factored out the GCF! But hold on, we're not done yet. We need to check if the expression inside the parentheses, $x^2 - 5x - 14$, can be factored further. This is where our next step comes in.

4. Factoring the Quadratic Expression

The expression inside the parentheses, $x^2 - 5x - 14$, is a quadratic expression. A quadratic expression is a polynomial of degree 2, meaning the highest power of the variable is 2. Factoring quadratic expressions is a common task in algebra, and there are several techniques we can use.

One common method is to look for two numbers that multiply to the constant term (-14) and add up to the coefficient of the linear term (-5). Let's call these two numbers 'a' and 'b'. We need to find 'a' and 'b' such that:

• a * b = -14
• a + b = -5

Let's think about the factors of -14. We have the following pairs: (1, -14), (-1, 14), (2, -7), and (-2, 7). Which of these pairs adds up to -5? Bingo! It's the pair (2, -7).

So, we can rewrite the quadratic expression $x^2 - 5x - 14$ as $(x + 2)(x - 7)$. This is the factored form of the quadratic expression.

5. The Completely Factored Form

Now, we're ready to put it all together! We factored out the GCF, $6x^2$, and then we factored the quadratic expression, $(x^2 - 5x - 14)$, into $(x + 2)(x - 7)$. So, the completely factored form of the polynomial $6x^4 - 30x^3 - 84x^2$ is:

$6x^2(x + 2)(x - 7)$

And that's it! We've successfully factored the polynomial completely. Give yourselves a pat on the back!

6. Checking Your Answer

It's always a good idea to check your answer, especially in math. To check our factored form, we can multiply the factors back together and see if we get the original polynomial. Let's do it:

$6x^2(x + 2)(x - 7)$

First, multiply the two binomials $(x + 2)$ and $(x - 7)$:

$(x + 2)(x - 7) = x^2 - 7x + 2x - 14 = x^2 - 5x - 14$

Now, multiply the result by $6x^2$:

$6x^2(x^2 - 5x - 14) = 6x^4 - 30x^3 - 84x^2$

Hey, look at that! We got back our original polynomial. This confirms that our factored form is correct.

7. Why This Answer Is the Right One

So, we've found that the completely factored form of $6x^4 - 30x^3 - 84x^2$ is $6x^2(x + 2)(x - 7)$. But why are the other options incorrect? Let's take a quick look.

• $6x^2(2x + 1)(7x - 1)$: If you multiply this out, you'll get a different polynomial than the original. The coefficients and signs won't match up.
• $6x^2(7x + 1)(2x - 1)$: Similar to the previous option, multiplying this out will not give you the original polynomial. The terms will be different.
• $6x^2(x - 2)(x + 7)$: This one is close, but the signs are flipped in the binomial factors. Multiplying it out will result in $6x^4 - 30x^3 + 84x^2$, which has a positive sign on the last term instead of a negative one.

Only our answer, $6x^2(x + 2)(x - 7)$, gives us the exact original polynomial when multiplied out. This is why it's the correct answer.

8. Tips and Tricks for Factoring

Factoring can become second nature with practice. Here are a few tips and tricks to help you along the way:

• Always look for the GCF first: This simplifies the polynomial and makes it easier to factor further.
• Know your factoring patterns: Common patterns like the difference of squares ($a^2 - b^2 = (a + b)(a - b)$) and perfect square trinomials ($a^2 + 2ab + b^2 = (a + b)^2$) can save you time.
• Practice, practice, practice: The more you factor, the better you'll become at recognizing patterns and applying the appropriate techniques.
• Don't be afraid to try different approaches: Sometimes, the first method you try might not work. Experiment with different techniques until you find one that does.
• Check your answers: Multiplying the factors back together is a great way to ensure you've factored correctly.

9. Conclusion: You've Got This!

Factoring polynomials is a crucial skill in algebra and beyond. By understanding the process and practicing regularly, you can master this technique and unlock new levels of mathematical problem-solving. We've successfully factored $6x^4 - 30x^3 - 84x^2$ into its completely factored form, $6x^2(x + 2)(x - 7)$.

Remember, guys, math is like a puzzle, and factoring is one of the tools you can use to solve it. Keep practicing, stay curious, and you'll be factoring polynomials like a pro in no time! Now go forth and conquer those algebraic expressions!