Factoring Polynomials Completely A Step-by-Step Guide

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Factoring polynomials is a fundamental skill in algebra, essential for solving equations, simplifying expressions, and understanding the behavior of functions. In this comprehensive guide, we will delve into the process of factoring the polynomial $12x^4 - 42x^3 - 90x^2$ completely. We will explore the concept of the Greatest Common Factor (GCF) and various factoring techniques to arrive at the fully factored form of the given polynomial. This guide aims to provide a clear and step-by-step approach, making it accessible for students and anyone looking to enhance their algebraic skills. Let's embark on this journey of polynomial factorization!

Understanding Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial expression into a product of simpler expressions, usually other polynomials or monomials. This is akin to breaking down a number into its prime factors. For instance, the number 12 can be factored into 2 × 2 × 3. Similarly, a polynomial like $x^2 + 5x + 6$ can be factored into $(x + 2)(x + 3)$. Factoring polynomials is a crucial skill in algebra as it simplifies complex expressions, helps in solving polynomial equations, and is used in various mathematical applications.

The importance of factoring polynomials stems from its wide-ranging applications in mathematics and other fields. In algebra, factoring is used to solve quadratic and higher-degree equations, which are essential in modeling real-world phenomena. Factoring simplifies expressions, making it easier to perform operations such as addition, subtraction, multiplication, and division. In calculus, factoring is used to find the roots of functions and to analyze their behavior. Moreover, polynomial factorization has applications in engineering, physics, computer science, and economics, where polynomials are used to model various systems and processes. Understanding the principles of polynomial factorization is therefore invaluable for anyone pursuing studies or careers in these fields.

Identifying the Greatest Common Factor (GCF)

Before diving into factoring a polynomial, it is essential to identify the Greatest Common Factor (GCF). The GCF is the largest factor that divides each term of the polynomial without leaving a remainder. Identifying the GCF is the first step in factoring a polynomial completely. It simplifies the polynomial and makes subsequent factoring steps easier.

To find the GCF, we look for the largest number that divides all the coefficients and the highest power of the variable that is common to all terms. For example, consider the polynomial $6x^3 + 9x^2 - 12x$. The GCF of the coefficients (6, 9, and -12) is 3. The highest power of x common to all terms is $x$. Therefore, the GCF of the polynomial is $3x$. Factoring out the GCF involves dividing each term of the polynomial by the GCF and writing the result as a product of the GCF and the remaining polynomial expression. This process simplifies the original polynomial and prepares it for further factorization if necessary.

Factoring $12x^4 - 42x^3 - 90x^2$: A Step-by-Step Guide

Step 1: Identify the GCF

Our polynomial is $12x^4 - 42x^3 - 90x^2$. To begin, we need to determine the Greatest Common Factor (GCF) of the coefficients and the variable terms.

First, let's find the GCF of the coefficients: 12, -42, and -90. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90. The largest number that divides all three coefficients is 6.

Next, we look at the variable terms: $x^4$, $x^3$, and $x^2$. The highest power of $x$ common to all terms is $x^2$. Therefore, the GCF of the polynomial is $6x^2$.

So, the GCF of $12x^4 - 42x^3 - 90x^2$ is $6x^2$.

Step 2: Factor out the GCF

Now that we have identified the GCF as $6x^2$, we factor it out from each term of the polynomial $12x^4 - 42x^3 - 90x^2$. This involves dividing each term by $6x^2$ and writing the polynomial as a product of the GCF and the resulting expression.

Dividing each term by $6x^2$:

• $12x^4 / 6x^2 = 2x^2$
• $-42x^3 / 6x^2 = -7x$
• $-90x^2 / 6x^2 = -15$

Thus, we can rewrite the polynomial as:

$12x^4 - 42x^3 - 90x^2 = 6x^2(2x^2 - 7x - 15)$

Factoring out the GCF simplifies the polynomial, making it easier to factor further. In this case, we have reduced the original polynomial to a product of the GCF, $6x^2$, and a quadratic expression, $2x^2 - 7x - 15$. The next step involves factoring this quadratic expression.

Step 3: Factor the Quadratic Expression

After factoring out the GCF, we are left with the quadratic expression $2x^2 - 7x - 15$. To factor this quadratic, we look for two binomials whose product equals the given expression. There are several methods to factor a quadratic expression, including trial and error, the quadratic formula, and factoring by grouping. Here, we will use the factoring by grouping method.

To factor $2x^2 - 7x - 15$ by grouping, we first look for two numbers that multiply to the product of the leading coefficient (2) and the constant term (-15), which is -30, and add up to the middle coefficient (-7). These two numbers are -10 and 3, since (-10) × 3 = -30 and (-10) + 3 = -7.

Next, we rewrite the middle term (-7x) using these two numbers:

$2x^2 - 7x - 15 = 2x^2 - 10x + 3x - 15$

Now, we group the terms in pairs:

$(2x^2 - 10x) + (3x - 15)$

We factor out the GCF from each pair:

$2x(x - 5) + 3(x - 5)$

Notice that both terms now have a common factor of $(x - 5)$. We factor out this common binomial:

$(2x + 3)(x - 5)$

Thus, the quadratic expression $2x^2 - 7x - 15$ factors into $(2x + 3)(x - 5)$.

Step 4: Write the Completely Factored Form

Having factored the quadratic expression, we can now write the completely factored form of the original polynomial. We combine the GCF we factored out in Step 2 with the factored quadratic expression from Step 3.

Recall that we factored out the GCF $6x^2$ from the original polynomial $12x^4 - 42x^3 - 90x^2$, resulting in $6x^2(2x^2 - 7x - 15)$. We then factored the quadratic expression $2x^2 - 7x - 15$ into $(2x + 3)(x - 5)$.

Therefore, the completely factored form of the polynomial $12x^4 - 42x^3 - 90x^2$ is:

$6x^2(2x + 3)(x - 5)$

This is the final factored form of the polynomial, where each factor cannot be factored further. This completes the process of factoring the polynomial completely.

Determining the GCF: The Correct Answer

In the process of factoring the polynomial $12x^4 - 42x^3 - 90x^2$, we identified the Greatest Common Factor (GCF) as the first crucial step. We examined the coefficients (12, -42, and -90) and the variable terms ($x^4$, $x^3$, and $x^2$) to determine the GCF.

We found that the GCF of the coefficients is 6, as it is the largest number that divides all three coefficients without leaving a remainder. For the variable terms, the highest power of $x$ common to all terms is $x^2$.

Therefore, the GCF of the polynomial $12x^4 - 42x^3 - 90x^2$ is $6x^2$.

Now, let's consider the given options:

• A. 3
• B. $3x$
• C. $6x$
• D. $6x^2$

Based on our analysis, the correct answer is:

• D. $6x^2$

This confirms that our step-by-step factoring process has led us to the correct identification of the GCF, which is a fundamental component of completely factoring the given polynomial.

Conclusion

In conclusion, we have successfully factored the polynomial $12x^4 - 42x^3 - 90x^2$ completely by following a systematic approach. The process involved several key steps, each crucial for arriving at the final factored form.

First, we identified the Greatest Common Factor (GCF) of the polynomial, which is $6x^2$. This step is essential as it simplifies the polynomial and makes subsequent factoring steps more manageable. We determined the GCF by examining the coefficients and the variable terms, ensuring we found the largest factor common to all terms.

Next, we factored out the GCF from the polynomial, resulting in the expression $6x^2(2x^2 - 7x - 15)$. This step reduced the original polynomial to a product of the GCF and a quadratic expression, setting the stage for further factorization.

We then factored the quadratic expression $2x^2 - 7x - 15$. This involved finding two binomials whose product equals the quadratic expression. We used the factoring by grouping method, which required finding two numbers that multiply to -30 (the product of the leading coefficient and the constant term) and add up to -7 (the middle coefficient). These numbers were -10 and 3, which allowed us to rewrite the quadratic and factor it into $(2x + 3)(x - 5)$.

Finally, we wrote the completely factored form of the polynomial by combining the GCF with the factored quadratic expression. This gave us the final factored form: $6x^2(2x + 3)(x - 5)$.

Throughout this process, we emphasized the importance of understanding each step and the underlying principles of polynomial factorization. This comprehensive guide aimed to provide a clear and step-by-step approach, making it accessible for students and anyone looking to enhance their algebraic skills. Factoring polynomials is a fundamental skill in algebra with wide-ranging applications in mathematics and other fields. By mastering this skill, one can solve complex equations, simplify expressions, and gain a deeper understanding of mathematical concepts.