Factoring 12x^4 + 6x^3 + 18x^2 A Step-by-Step Guide

Factoring polynomials is a fundamental skill in algebra. In this comprehensive guide, we will delve into the process of factoring the polynomial $12x^4 + 6x^3 + 18x^2$ completely. This involves identifying the greatest common factor (GCF) and expressing the polynomial as a product of simpler factors. Understanding factoring is crucial for solving equations, simplifying expressions, and tackling more advanced mathematical concepts. This article aims to provide a step-by-step approach to factoring this specific polynomial, while also highlighting the underlying principles and techniques applicable to a wide range of factoring problems.

Understanding the Basics of Factoring

Before diving into the specifics of factoring our polynomial, let's revisit the basic concept of factoring. Factoring is essentially the reverse process of expanding. When we expand an expression, we multiply terms together to obtain a polynomial. Factoring, on the other hand, involves breaking down a polynomial into its constituent factors – expressions that, when multiplied together, produce the original polynomial. The ability to factor effectively is a cornerstone of algebraic manipulation and problem-solving. It allows us to simplify complex expressions, solve equations, and gain deeper insights into the structure of mathematical relationships.

To illustrate, consider the simple expression $2x + 4$. We can factor out a common factor of 2, resulting in $2(x + 2)$. Here, 2 and $(x + 2)$ are the factors of the original expression. The goal of factoring is to decompose a given polynomial into its simplest factors, often referred to as prime factors. This process involves identifying common factors among the terms of the polynomial and expressing it as a product of these factors.

Factoring polynomials often involves recognizing patterns and applying specific techniques. Some common factoring techniques include:

• Greatest Common Factor (GCF): Identifying the largest factor that divides all terms of the polynomial.
• Difference of Squares: Factoring expressions of the form $a^2 - b^2$ as $(a + b)(a - b)$.
• Perfect Square Trinomials: Factoring expressions of the form $a^2 + 2ab + b^2$ as $(a + b)^2$ or $a^2 - 2ab + b^2$ as $(a - b)^2$.
• Factoring by Grouping: Grouping terms in a polynomial to identify common factors.
• Trial and Error: Systematically testing different combinations of factors.

In the case of $12x^4 + 6x^3 + 18x^2$, we will primarily focus on the greatest common factor (GCF) technique, as it is the most straightforward approach for this particular polynomial. Understanding these fundamental concepts and techniques is crucial for mastering factoring and its applications in algebra and beyond. The process of factoring not only simplifies expressions but also reveals underlying relationships and structures within mathematical equations, making it an invaluable tool for problem-solving and mathematical reasoning.

Identifying the Greatest Common Factor (GCF)

In factoring the polynomial $12x^4 + 6x^3 + 18x^2$, the first and most crucial step is to identify the greatest common factor (GCF) of all the terms. The GCF is the largest factor that divides each term of the polynomial without leaving a remainder. This factor can be a number, a variable, or a combination of both. Finding the GCF simplifies the polynomial and makes it easier to factor further.

To determine the GCF, we first look at the coefficients (the numerical parts) of the terms: 12, 6, and 18. We need to find the largest number that divides all three of these coefficients. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 6 are 1, 2, 3, and 6. The factors of 18 are 1, 2, 3, 6, 9, and 18. By comparing these factors, we can see that the greatest common factor of 12, 6, and 18 is 6.

Next, we consider the variable parts of the terms: $x^4$, $x^3$, and $x^2$. The GCF for the variables is the variable raised to the lowest power present in the terms. In this case, the lowest power of $x$ is $x^2$. Therefore, the GCF for the variable parts is $x^2$.

Combining the numerical GCF and the variable GCF, we find that the overall GCF of $12x^4$, $6x^3$, and $18x^2$ is $6x^2$. This means that $6x^2$ is the largest expression that can divide each term of the polynomial without leaving a remainder. Recognizing and extracting the GCF is a critical step in factoring, as it not only simplifies the polynomial but also reveals its underlying structure. By factoring out the GCF, we are essentially reversing the distributive property, which is a fundamental concept in algebra. This process allows us to express the polynomial as a product of simpler factors, making it easier to analyze and manipulate.

Understanding how to identify the GCF is essential for mastering factoring. It lays the foundation for more complex factoring techniques and problem-solving strategies in algebra. By systematically breaking down the polynomial and identifying the common factors, we can simplify expressions and reveal the inherent mathematical relationships within them. This skill is not only crucial for academic success but also has practical applications in various fields, such as engineering, computer science, and economics, where mathematical modeling and simplification are essential.

Factoring out the GCF

Now that we have identified the greatest common factor, $6x^2$, we can proceed with factoring it out of the polynomial $12x^4 + 6x^3 + 18x^2$. Factoring out the GCF involves dividing each term of the polynomial by the GCF and writing the result in a factored form. This process is essentially the reverse of the distributive property, allowing us to express the polynomial as a product of the GCF and a simpler polynomial.

To factor out $6x^2$, we divide each term of the polynomial by $6x^2$:

• $12x^4 / (6x^2) = 2x^2$
• $6x^3 / (6x^2) = x$
• $18x^2 / (6x^2) = 3$

These results become the terms inside the parentheses when we write the factored form. The GCF, $6x^2$, is placed outside the parentheses as the common factor. Therefore, the factored form of the polynomial is:

$6x^2(2x^2 + x + 3)$

This expression indicates that the original polynomial $12x^4 + 6x^3 + 18x^2$ can be expressed as the product of $6x^2$ and the trinomial $(2x^2 + x + 3)$. Factoring out the GCF is a crucial step in simplifying polynomials and is often the first step in more complex factoring problems. By extracting the common factor, we reduce the degree and complexity of the remaining polynomial, making it easier to analyze and potentially factor further.

The process of factoring out the GCF not only simplifies the polynomial but also reveals its underlying structure. It allows us to see the polynomial as a product of simpler expressions, which can be beneficial for solving equations, finding roots, and understanding the behavior of the polynomial function. This technique is a fundamental tool in algebra and is essential for mastering more advanced concepts in mathematics. By consistently applying the GCF factoring method, students can develop a strong foundation in algebraic manipulation and problem-solving.

Furthermore, the ability to factor out the GCF has practical applications in various fields beyond mathematics. In engineering, for example, simplifying expressions is crucial for designing and analyzing systems. In computer science, factoring can be used to optimize algorithms and reduce computational complexity. The skill of factoring, therefore, is not only valuable in academic settings but also in real-world applications where mathematical simplification is essential.

Checking for Further Factoring

After factoring out the GCF, it is crucial to check whether the remaining polynomial can be factored further. This step ensures that the polynomial is factored completely, meaning it is expressed as a product of prime factors or irreducible polynomials. An irreducible polynomial is one that cannot be factored into polynomials of lower degree with coefficients from the same set (e.g., integers). Checking for further factoring is a vital part of the factoring process, as it guarantees the most simplified and complete form of the factored expression.

In our case, after factoring out the GCF $6x^2$, we are left with the trinomial $2x^2 + x + 3$. To determine if this trinomial can be factored further, we can explore several techniques. One common method is to attempt to factor the trinomial into two binomials of the form $(ax + b)(cx + d)$. This involves finding two numbers that multiply to give the product of the leading coefficient (2) and the constant term (3), which is 6, and add up to the middle coefficient (1).

Let's consider the factors of 6: 1 and 6, 2 and 3. We need to find a pair of factors that can be combined to give a sum of 1. However, no combination of these factors adds up to 1. This suggests that the trinomial $2x^2 + x + 3$ may not be factorable using integer coefficients. Another way to confirm this is by using the discriminant.

The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is given by the formula $D = b^2 - 4ac$. If the discriminant is negative, the quadratic equation has no real roots, and the trinomial cannot be factored into linear factors with real coefficients. If the discriminant is a perfect square, the quadratic can be factored into two binomials with integer coefficients. If the discriminant is positive but not a perfect square, the quadratic has real roots but cannot be factored into binomials with integer coefficients.

For our trinomial $2x^2 + x + 3$, $a = 2$, $b = 1$, and $c = 3$. The discriminant is:

$D = 1^2 - 4(2)(3) = 1 - 24 = -23$

Since the discriminant is negative (-23), the trinomial $2x^2 + x + 3$ cannot be factored further using real numbers. This confirms that the factored form $6x^2(2x^2 + x + 3)$ is indeed the complete factorization of the original polynomial $12x^4 + 6x^3 + 18x^2$.

This step of checking for further factoring is not just a procedural necessity but also a critical thinking exercise in algebra. It reinforces the understanding of factoring principles and techniques and enhances problem-solving skills. By systematically examining the remaining polynomial, students can develop a deeper appreciation for the structure and properties of algebraic expressions.

Final Answer

After a thorough step-by-step process of factoring the polynomial $12x^4 + 6x^3 + 18x^2$, we have arrived at the completely factored form. We began by identifying the greatest common factor (GCF) of the terms, which was determined to be $6x^2$. We then factored out the GCF from the polynomial, resulting in the expression $6x^2(2x^2 + x + 3)$. Finally, we checked whether the remaining trinomial, $2x^2 + x + 3$, could be factored further.

Using the discriminant method, we found that the discriminant of the trinomial is -23, which is a negative value. This indicates that the trinomial cannot be factored further using real numbers. Therefore, the polynomial is completely factored as $6x^2(2x^2 + x + 3)$. This final factored form represents the original polynomial as a product of its simplest factors, satisfying the requirement of complete factorization.

The process of factoring polynomials is a fundamental skill in algebra, with applications in various areas of mathematics and beyond. It involves identifying common factors, applying factoring techniques, and verifying that the polynomial is factored completely. This comprehensive guide has demonstrated the step-by-step approach to factoring a specific polynomial, highlighting the key concepts and techniques involved. Understanding and mastering these factoring skills is essential for success in algebra and higher-level mathematics.

Therefore, the complete factorization of $12x^4 + 6x^3 + 18x^2$ is $6x^2(2x^2 + x + 3)$.

Given the options:

A. Prime B. $6(2x^4 + x^3 + 3x^2)$ C. $6x^2(2x^2 + x + 3)$ D. $6x(2x^3 + x^2 + 3x)$

The correct answer is C. $6x^2(2x^2 + x + 3)$. This is because it represents the polynomial factored completely, with the GCF properly factored out and the remaining trinomial shown to be irreducible over real numbers. The other options either do not factor out the GCF completely (option D) or do not factor the expression at all (options A and B).