Factored Form Of 6n^4 - 24n^3 + 18n: A Step-by-Step Guide

Aug 4, 2025 by Felix Dubois 58 views

Factoring Polynomials: A Step-by-Step Guide to $6n^4 - 24n^3 + 18n$

Hey guys! Let's dive into the world of factoring polynomials. Today, we’re tackling a specific problem: finding the factored form of the expression $6n^4 - 24n^3 + 18n$ . Factoring polynomials might seem daunting at first, but with a systematic approach, it becomes a breeze. We’ll break down each step, ensuring you not only understand the solution but also the underlying principles. So, grab your pencils, and let's get started!

Understanding Factoring

Before we jump into the problem, let's quickly recap what factoring is all about. In simple terms, factoring is like reverse multiplication. Think of it this way: when you multiply two numbers (or expressions) together, you get a product. Factoring is the process of breaking down that product back into its original factors. For polynomials, this means expressing a complex expression as a product of simpler expressions.

Why is factoring so important? Well, it's a fundamental skill in algebra and calculus. Factoring helps us simplify expressions, solve equations, and analyze functions. From finding the roots of a polynomial to simplifying rational expressions, factoring is a versatile tool in your mathematical arsenal.

When we talk about the "factored form" of a polynomial, we mean expressing it as a product of its factors. This usually involves identifying common factors, using factoring patterns, or applying other algebraic techniques. The goal is to break down the polynomial into its simplest components, making it easier to work with.

Now, let's apply these concepts to our specific problem. We'll start by identifying the greatest common factor (GCF) and then proceed to further factor the expression, step by step. Remember, practice makes perfect, so the more you factor, the more comfortable you'll become with the process.

Step 1: Identify 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 terms of the polynomial. It's like finding the biggest piece of the puzzle that fits into all the other pieces. Identifying and factoring out the GCF simplifies the polynomial, making subsequent factoring steps much easier.

In our expression, $6n^4 - 24n^3 + 18n$ , we need to find the GCF of the coefficients (6, -24, and 18) and the variable terms ( $n^4$ , $n^3$ , and $n$ ). Let’s start with the coefficients. The factors of 6 are 1, 2, 3, and 6. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 18 are 1, 2, 3, 6, 9, and 18. The largest number that appears in all three lists is 6. So, the GCF of the coefficients is 6.

Now, let’s look at the variable terms. We have $n^4$ , $n^3$ , and $n$ . The variable $n$ appears in all three terms. To find the GCF of the variable terms, we take the lowest power of $n$ that appears in the expression. In this case, it’s $n^1$ (or simply $n$ ).

Combining the GCF of the coefficients and the variable terms, we find that the GCF of the entire polynomial $6n^4 - 24n^3 + 18n$ is $6n$ . This means we can factor out $6n$ from each term in the expression. Factoring out the GCF is a crucial step because it reduces the complexity of the polynomial and often reveals further factoring opportunities. So, always start by looking for the GCF – it's your best friend in the factoring process!

Step 2: Factor out the GCF

Now that we’ve identified the greatest common factor (GCF) as $6n$ , the next step is to factor it out from the polynomial. Factoring out the GCF is like pulling out the common thread that runs through all the terms, leaving behind a simpler expression inside the parentheses. This process involves dividing each term of the polynomial by the GCF and writing the result inside the parentheses.

Our polynomial is $6n^4 - 24n^3 + 18n$ . To factor out $6n$ , we’ll divide each term by $6n$ :

$6n^4 / 6n = n^3$
$-24n^3 / 6n = -4n^2$
$18n / 6n = 3$

So, when we divide each term by $6n$ , we get $n^3$ , $-4n^2$ , and $3$ . Now, we write these terms inside the parentheses, with the GCF, $6n$ , outside:

$6n(n^3 - 4n^2 + 3)$

This is a significant step forward! We’ve successfully factored out the GCF, which simplifies our original expression. What we have now is $6n$ multiplied by a cubic polynomial ( $n^3 - 4n^2 + 3$ ). The expression inside the parentheses is now much easier to work with, and we can proceed to the next step: checking if the expression inside the parentheses can be factored further.

Remember, factoring out the GCF is a fundamental technique in polynomial factoring. It not only simplifies the expression but also sets the stage for further factoring steps. Always look for the GCF as the first step in any factoring problem – it’s a game-changer!

Step 3: Check for Further Factoring

After factoring out the greatest common factor (GCF), the next crucial step is to check if the remaining expression can be factored further. This is where your detective skills come into play! You need to examine the expression inside the parentheses and see if it fits any common factoring patterns or if there are any other factors lurking within.

In our case, we have the expression $6n(n^3 - 4n^2 + 3)$ . We’ve already factored out the $6n$ , so now we need to focus on the cubic polynomial inside the parentheses: $n^3 - 4n^2 + 3$ .

Cubic polynomials can be a bit tricky to factor, but there are a few strategies we can try. One common approach is to look for rational roots using the Rational Root Theorem. Another approach is to try factoring by grouping, but this often works best when there are four terms in the polynomial.

In this particular case, we can try a simple approach first: see if there are any obvious factors by testing small integer values for $n$ . For example, let’s try $n = 1$ :

$(1)^3 - 4(1)^2 + 3 = 1 - 4 + 3 = 0$

Great! We found that $n = 1$ is a root of the polynomial. This means that $(n - 1)$ is a factor. Now, we can use polynomial long division or synthetic division to divide $n^3 - 4n^2 + 3$ by $(n - 1)$ to find the other factor.

This step is all about exploration and applying different factoring techniques. Sometimes, the expression might not be factorable, and that’s perfectly okay. But always make sure to check for further factoring opportunities – you might just uncover the final pieces of the puzzle!

Step 4: Polynomial Long Division (or Synthetic Division)

Since we found that $(n - 1)$ is a factor of the cubic polynomial $n^3 - 4n^2 + 3$ , we need to divide the polynomial by $(n - 1)$ to find the remaining quadratic factor. We can use either polynomial long division or synthetic division for this. Both methods achieve the same result, but synthetic division is often quicker and more efficient for linear divisors like $(n - 1)$ . Let’s use synthetic division for this example.

To set up synthetic division, we write the coefficients of the polynomial (1, -4, 0, 3) and the root (1) as follows:

1 | 1 -4 0 3
 |
 -----------------

Now, we perform the synthetic division:

Bring down the first coefficient (1):

1 | 1 -4 0 3
 | 1
 -----------------
 1

Multiply the root (1) by the number we just brought down (1) and write the result under the next coefficient (-4):

1 | 1 -4 0 3
 | 1
 -----------------
 1

Add the numbers in the second column (-4 and 1):

1 | 1 -4 0 3
 | 1
 -----------------
 1 -3

Multiply the root (1) by the result (-3) and write it under the next coefficient (0):

1 | 1 -4 0 3
 | 1 -3
 -----------------
 1 -3

Add the numbers in the third column (0 and -3):

1 | 1 -4 0 3
 | 1 -3
 -----------------
 1 -3 -3

Multiply the root (1) by the result (-3) and write it under the last coefficient (3):

1 | 1 -4 0 3
 | 1 -3 -3
 -----------------
 1 -3 -3

Add the numbers in the last column (3 and -3):

1 | 1 -4 0 3
 | 1 -3 -3
 -----------------
 1 -3 -3 0

The last number (0) is the remainder, and the other numbers (1, -3, -3) are the coefficients of the quotient. So, the quotient is $n^2 - 3n - 3$ .

This means that $n^3 - 4n^2 + 3 = (n - 1)(n^2 - 3n - 3)$ . We’ve successfully divided the cubic polynomial and found its factors!

Polynomial long division (or synthetic division) is a powerful technique for factoring higher-degree polynomials. It allows us to break down complex expressions into simpler factors, making them easier to analyze and solve.

Step 5: Factor the Quadratic (If Possible)

After dividing the cubic polynomial by $(n - 1)$ , we obtained the quadratic factor $n^2 - 3n - 3$ . Now, the next step is to determine if this quadratic can be factored further. Factoring quadratics is a common skill in algebra, and there are several techniques we can use, such as looking for two numbers that multiply to the constant term and add up to the coefficient of the linear term, or using the quadratic formula.

In our case, we have $n^2 - 3n - 3$ . We need to find two numbers that multiply to -3 and add up to -3. Let’s list the factors of -3:

-1 and 3
1 and -3

Neither of these pairs adds up to -3. This means that the quadratic cannot be factored using simple integer factors. In this situation, we can use the quadratic formula to find the roots, but since the question asks for the factored form, and we cannot factor it further using integers, we leave the quadratic as it is.

The quadratic formula is a powerful tool for finding the roots of any quadratic equation, but sometimes, the roots are irrational or complex numbers. In such cases, the quadratic cannot be factored into linear factors with integer coefficients.

This step is a crucial checkpoint in the factoring process. Not all quadratics can be factored easily, and recognizing when to stop is just as important as knowing how to factor. In our case, we’ve determined that $n^2 - 3n - 3$ is irreducible over the integers, so we move on to the final step.

Step 6: Write the Final Factored Form

Finally, we’ve reached the last step! Now, we need to write out the complete factored form of the original polynomial. This involves combining all the factors we’ve identified in the previous steps. It’s like putting all the pieces of the puzzle back together to see the complete picture.

We started with the polynomial $6n^4 - 24n^3 + 18n$ . We first factored out the greatest common factor (GCF), $6n$ , which gave us:

$6n(n^3 - 4n^2 + 3)$

Then, we found that $(n - 1)$ is a factor of the cubic polynomial $n^3 - 4n^2 + 3$ . After dividing the cubic polynomial by $(n - 1)$ , we obtained the quadratic factor $n^2 - 3n - 3$ . We determined that this quadratic cannot be factored further using integers.

So, the final factored form of the polynomial is:

$6n(n - 1)(n^2 - 3n - 3)$

This is the complete factored form of the polynomial. We’ve broken down the original expression into its simplest factors, and we can’t factor it any further using integer coefficients.

Writing the final factored form is the culmination of all our efforts. It’s the moment where we see the result of our hard work and the elegance of the factored expression. Always double-check your work to make sure you haven’t missed any factors and that the final factored form is indeed the simplest possible expression.

Conclusion

Great job, guys! We’ve successfully factored the polynomial $6n^4 - 24n^3 + 18n$ into its factored form: $6n(n - 1)(n^2 - 3n - 3)$ . We walked through each step of the process, from identifying the greatest common factor (GCF) to using polynomial division and checking for further factoring opportunities. Factoring polynomials can be challenging, but with a systematic approach and plenty of practice, you’ll become a factoring pro in no time!

Remember, the key to mastering factoring is to break down the problem into smaller, manageable steps. Always start by looking for the GCF, then check for common factoring patterns, and don’t be afraid to use techniques like polynomial division when necessary. Keep practicing, and you’ll find that factoring becomes second nature. Keep up the awesome work, and happy factoring!