Factored Form Of 6n⁴ - 24n³ + 18n A Step-by-Step Solution
Hey guys! Today, we're diving into a fun math problem: finding the factored form of the polynomial expression 6n⁴ - 24n³ + 18n. Factoring polynomials is a crucial skill in algebra, and it's super useful for solving equations, simplifying expressions, and even tackling more advanced math concepts later on. So, let's break it down step by step and make sure we understand exactly how to get to the correct factored form. Let’s solve this step by step.
Understanding Factoring
Before we jump into the problem itself, let’s quickly recap what factoring is all about. In simple terms, factoring is like the reverse of expanding. When we expand, we multiply things out; when we factor, we break things down into their multiplicative components. Think of it like this: if you have the number 12, you can factor it into 2 × 6 or 3 × 4. Similarly, with polynomials, we're looking for expressions that, when multiplied together, give us the original polynomial.
In this particular problem, our main goal is to express the given polynomial, 6n⁴ - 24n³ + 18n, as a product of simpler factors. This not only makes the expression easier to work with but also provides insights into the polynomial’s roots and behavior. Factoring is a cornerstone of algebraic manipulation and equation-solving, making it an indispensable skill for anyone venturing further into mathematics.
The benefits of mastering factoring extend beyond mere academic exercises. Factored forms of expressions can reveal hidden symmetries, simplify complex calculations, and even offer practical advantages in fields like engineering, computer science, and economics. For instance, in computer science, factored polynomials can help optimize algorithms and reduce computational complexity. In engineering, they might be used to analyze the stability of systems or to design efficient structures. The ability to factor efficiently is therefore not just an abstract mathematical skill but a powerful tool with wide-ranging applications.
To approach any factoring problem, it’s essential to have a systematic strategy. Start by identifying common factors among the terms. Look for both numerical and variable factors. In our case, we’ll see that all terms share a common factor involving both a number and a variable. Once you’ve factored out the greatest common factor (GCF), examine the remaining expression. It might be a quadratic that can be factored using techniques like looking for factor pairs or using the quadratic formula. Or, it could be another type of polynomial that requires more advanced methods, such as grouping or recognizing special patterns like the difference of squares or sum/difference of cubes. Practice and familiarity with these techniques are key to becoming proficient at factoring.
Step-by-Step Factoring of 6n⁴ - 24n³ + 18n
1. Identify the Greatest Common Factor (GCF)
The first step in factoring any polynomial is to look for the Greatest Common Factor (GCF). This is the largest term that divides evenly into all terms of the polynomial. In our expression, 6n⁴ - 24n³ + 18n, we need to find the GCF of the coefficients (6, -24, and 18) and the variable terms (n⁴, n³, 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 numerical GCF is 6.
Now, let’s look at the variable terms: n⁴, n³, and n. The GCF here is the lowest power of n present in all terms, which is n (or n¹). Therefore, the variable GCF is n.
Combining these, the Greatest Common Factor (GCF) for the entire polynomial is 6n. Factoring out the GCF simplifies the polynomial, making subsequent steps easier to manage. This initial step is crucial because it reduces the complexity of the expression and sets the stage for further factorization, if needed. By pulling out the GCF, we're essentially stripping away the most obvious commonality, allowing us to focus on the underlying structure of the remaining expression. It’s a bit like peeling back the layers of an onion to get to its core.
2. Factor Out the GCF
Now that we've identified the GCF as 6n, we factor it out from each term in the polynomial: 6n⁴ - 24n³ + 18n. This means we divide each term by 6n and write the result inside a set of parentheses, with 6n outside.
- 6n⁴ ÷ 6n = n³
- -24n³ ÷ 6n = -4n²
- 18n ÷ 6n = 3
So, when we factor out 6n, we get: 6n(n³ - 4n² + 3). This step is vital because it simplifies the original expression into a more manageable form. Factoring out the GCF is akin to finding the lowest common denominator in fractions—it streamlines the expression, making it easier to perform further operations. The result, 6n(n³ - 4n² + 3), shows the GCF outside and the remaining polynomial inside the parentheses.
This factored form not only simplifies the expression but also provides insights into its structure. For instance, we can immediately see that n = 0 is a root of the polynomial, since setting n = 0 makes the entire expression zero. Further analysis of the cubic polynomial inside the parentheses may reveal additional roots or factors. The process of factoring out the GCF is a fundamental technique that lays the groundwork for more advanced algebraic manipulations, such as solving equations and simplifying rational expressions. It’s like the foundation of a building—essential for the structure to stand firm.
3. Check the Remaining Polynomial for Further Factoring
After factoring out the GCF, we're left with the expression inside the parentheses: (n³ - 4n² + 3). Now, we need to check if this polynomial can be factored further. Factoring isn't always a one-step process; sometimes, you need to apply different techniques to fully break down an expression.
The cubic polynomial (n³ - 4n² + 3) doesn’t fit any immediate patterns like the difference of squares or cubes, so we need to explore other methods. One common approach for factoring cubics is to look for rational roots using the Rational Root Theorem. This theorem suggests that any rational root of the polynomial will be a factor of the constant term (in this case, 3) divided by a factor of the leading coefficient (which is 1). So, the possible rational roots are ±1 and ±3.
We can test these potential roots by substituting them into the polynomial to see if they make the expression equal to zero. Let’s start with n = 1:
(1)³ - 4(1)² + 3 = 1 - 4 + 3 = 0
Since substituting n = 1 gives us zero, it means that (n - 1) is a factor of the cubic polynomial. This is a crucial discovery because it allows us to reduce the cubic to a quadratic, which is often easier to factor.
The technique of checking for further factoring is a hallmark of thorough algebraic problem-solving. It ensures that the expression is broken down into its simplest possible components. Just as a detective follows every lead, a mathematician explores every avenue of factorization. The pursuit of complete factorization is not merely about obtaining a final answer; it’s about gaining a deeper understanding of the underlying mathematical structure.
4. Use Synthetic Division or Polynomial Long Division
Since we found that (n - 1) is a factor of n³ - 4n² + 3, we can use synthetic division or polynomial long division to find the remaining quadratic factor. These techniques help us divide the cubic polynomial by the linear factor (n - 1).
Let’s use synthetic division:
1 | 1 -4 0 3
| 1 -3 -3
----------------
1 -3 -3 0
The numbers in the bottom row (1, -3, -3) represent the coefficients of the quotient, which is a quadratic polynomial. The last number (0) confirms that the remainder is zero, meaning that (n - 1) divides evenly into the cubic polynomial. The quotient is n² - 3n - 3.
So, n³ - 4n² + 3 can be written as (n - 1)(n² - 3n - 3). This step is a pivotal moment in our factorization journey. We’ve transformed a challenging cubic expression into the product of a linear and a quadratic factor. The use of synthetic division (or polynomial long division) is a powerful technique for simplifying higher-degree polynomials once a factor has been identified. It’s a bit like using a key to unlock a door, revealing the next part of the problem.
5. Analyze the Quadratic Factor
Now we have 6n(n - 1)(n² - 3n - 3). The next step is to analyze the quadratic factor, n² - 3n - 3, to see if it can be factored further. To do this, we can try to find two numbers that multiply to -3 and add to -3. If we can find such numbers, we can factor the quadratic into two binomials.
However, in this case, it's not immediately obvious what those numbers might be. This is where we can use the discriminant, which is part of the quadratic formula, to determine whether the quadratic has real roots (and thus can be factored over real numbers). The discriminant is given by the formula:
Δ = b² - 4ac
For the quadratic n² - 3n - 3, a = 1, b = -3, and c = -3. Plugging these values into the discriminant formula, we get:
Δ = (-3)² - 4(1)(-3) = 9 + 12 = 21
Since the discriminant (21) is positive and not a perfect square, the quadratic equation n² - 3n - 3 = 0 has two distinct real roots, but they are irrational. This means that the quadratic cannot be factored further using simple integer or fractional coefficients. We would need to use the quadratic formula to find the exact roots, but for the purpose of factoring over integers, we’ve gone as far as we can.
This analysis of the quadratic factor is a critical step in the factoring process. It’s akin to a quality control check, ensuring that we don’t waste time trying to factor something that is inherently unfactorable using elementary methods. The discriminant acts as a diagnostic tool, guiding us to the most efficient path forward. In this case, it tells us that the quadratic n² - 3n - 3 is irreducible over integers, and we can move on with our final factored form.
6. Write the Final Factored Form
After our step-by-step analysis, we've determined that the polynomial 6n⁴ - 24n³ + 18n can be factored as:
6n(n - 1)(n² - 3n - 3)
This is the final factored form of the expression. We started by pulling out the GCF, then factored the resulting cubic polynomial by finding a root and using synthetic division. Finally, we analyzed the resulting quadratic factor and determined that it could not be factored further over integers. Therefore, we’ve arrived at the most simplified factored expression.
Conclusion
So, the factored form of 6n⁴ - 24n³ + 18n is 6n(n - 1)(n² - 3n - 3). This matches option C. 6n(n³ - 4n² + 3), when you partially factor it by taking 6n out. Factoring polynomials can seem tricky at first, but by following these steps – identifying the GCF, looking for rational roots, and using techniques like synthetic division – you'll become a pro in no time. Keep practicing, and you'll find that factoring becomes second nature. And remember, math can be fun when you break it down step by step. Keep up the great work, guys!