What Did Faelyn Notice And What Is The Next Step In Factoring The Polynomial 6x^4 - 8x^2 + 3x^2 + 4 After Grouping And Factoring The GCF?

Factoring polynomials can sometimes feel like solving a puzzle, and one common technique is factoring by grouping. This method is particularly useful when dealing with polynomials with four or more terms. In this article, we will delve into the process of factoring by grouping, using an example where Faelyn attempts to factor the polynomial ${6x^4 - 8x^2 + 3x^2 - 4}$. We'll break down each step, identify potential errors, and provide a comprehensive explanation to ensure a clear understanding of the correct approach.

Faelyn's Attempt

Faelyn's attempt to factor the polynomial ${6x^4 - 8x^2 + 3x^2 - 4}$ involves grouping terms and factoring out the greatest common factor (GCF) from each group. Her work is presented in the following steps:

Step 1: Faelyn groups the terms as follows:

${(6x^4 - 8x^2) + (3x^2 - 4)}$

Step 2: She factors out the GCF from each group:

${2x^2(3x^2 - 4) + 1(3x^2 - 4)}$

Analyzing Faelyn's Steps

Let's analyze each step to see if Faelyn is on the right track.

Step 1 involves grouping terms, which is a standard initial move in factoring by grouping. The grouping itself is correct; she has simply placed parentheses around the first two terms and the last two terms. This step sets the stage for identifying common factors within each group.

In Step 2, Faelyn factors out the greatest common factor (GCF) from each group. For the first group, ${6x^4 - 8x^2}$, the GCF is ${2x^2}$. Factoring this out, she correctly gets ${2x^2(3x^2 - 4)}$. For the second group, ${3x^2 - 4}$, the GCF is 1, so factoring it out gives ${1(3x^2 - 4)}$. So far, Faelyn's work is accurate.

The Next Step Factoring by Grouping

After factoring out the GCF from each group, the next crucial step in factoring by grouping is to identify whether the expressions inside the parentheses are the same. If they are, we can factor out the common binomial. Looking at Faelyn's result:

${2x^2(3x^2 - 4) + 1(3x^2 - 4)}$

We can see that the binomial ${(3x^2 - 4)}$ is common to both terms. This allows us to factor it out, which is the essence of the grouping method. Factoring out ${(3x^2 - 4)}$ from the entire expression, we get:

${(3x^2 - 4)(2x^2 + 1)}$

This is the completely factored form of the original polynomial.

Correcting the Polynomial: A Crucial Observation

There seems to be a slight error in the original presentation of the polynomial. It is written as ${6x^4 - 8x^2 + 3x^2 + 4}$. If this is the case, the constant term should be -4 and not +4 for the factoring by grouping method to work as intended in Faelyn's steps. If the original polynomial was indeed ${6x^4 - 8x^2 + 3x^2 - 4}$, Faelyn's grouping and factoring would lead to the correct result, as we've shown above.

However, if the polynomial is ${6x^4 - 8x^2 + 3x^2 + 4}$, the process would be different. First, combine the like terms:

${6x^4 - 5x^2 + 4}$

Now, we attempt to factor this quadratic form. We look for two numbers that multiply to ${6 imes 4 = 24}$ and add up to -5. There are no such integer numbers, which means this polynomial cannot be factored using simple integer coefficients.

Therefore, for the factoring by grouping method to work as demonstrated in Faelyn's steps, the original polynomial should be ${6x^4 - 8x^2 + 3x^2 - 4}$.

Step-by-Step Solution with the Corrected Polynomial

Let's assume the original polynomial was ${6x^4 - 8x^2 + 3x^2 - 4}$. We will go through the factoring by grouping process step by step.

1. Group the terms:

${(6x^4 - 8x^2) + (3x^2 - 4)}$

This involves grouping the first two terms and the last two terms together. This is a standard starting point for factoring by grouping.

2. Factor out the GCF from each group:

From the first group, ${6x^4 - 8x^2}$, the GCF is ${2x^2}$. Factoring this out, we get:

${2x^2(3x^2 - 4)}$

From the second group, ${3x^2 - 4}$, the GCF is 1. Factoring this out, we get:

${1(3x^2 - 4)}$

Combining these, we have:

${2x^2(3x^2 - 4) + 1(3x^2 - 4)}$

3. Factor out the common binomial:

We notice that ${(3x^2 - 4)}$ is a common binomial factor. Factoring this out, we get:

${(3x^2 - 4)(2x^2 + 1)}$

This is the completely factored form of the polynomial ${6x^4 - 8x^2 + 3x^2 - 4}$.

Common Mistakes and How to Avoid Them

When factoring by grouping, several common mistakes can occur. Being aware of these pitfalls can help prevent errors.

Mistake 1 Incorrectly Identifying the GCF

One common mistake is incorrectly identifying the greatest common factor (GCF). For example, in the term ${6x^4 - 8x^2}$, some might mistakenly factor out only ${x^2}$ instead of ${2x^2}$. Always ensure you are factoring out the largest possible factor, both in terms of coefficients and variables.

To avoid this, break down each term into its prime factors and identify the common factors. For ${6x^4}$, the prime factors are ${2 imes 3 imes x imes x imes x imes x}$, and for ${8x^2}$, the prime factors are ${2 imes 2 imes 2 imes x imes x}$. The common factors are ${2 imes x imes x}$, which gives us the GCF ${2x^2}$.

Mistake 2 Sign Errors

Sign errors are another frequent issue. When factoring out a negative GCF, ensure you change the signs of the remaining terms inside the parentheses correctly. For example, if you have ${-4x - 8}$ and factor out -4, you should get ${-4(x + 2)}$, not ${-4(x - 2)}$.

To prevent sign errors, double-check your distribution after factoring. Ensure that multiplying the GCF back into the parentheses yields the original expression.

Mistake 3 Not Grouping Correctly

Sometimes, the initial grouping might not lead to a common binomial factor. In such cases, try rearranging the terms and grouping differently. For instance, if ${(ax + by) + (ay + bx)}$ doesn't immediately reveal a common factor, rearranging to ${(ax + bx) + (ay + by)}$ might make the common factors more apparent.

Mistake 4 Stopping Too Early

A common error is stopping the factoring process before the polynomial is fully factored. After factoring by grouping, always check if the resulting binomial factors can be factored further. For example, if you end up with ${(x^2 - 4)(x + 2)}$, recognize that ${(x^2 - 4)}$ is a difference of squares and can be further factored into ${(x - 2)(x + 2)}$. Thus, the fully factored form would be ${(x - 2)(x + 2)(x + 2)}$.

Mistake 5 Assuming All Polynomials Can Be Factored

Not all polynomials can be factored using simple methods or integer coefficients. If you've tried grouping, rearranging, and other techniques without success, the polynomial might be irreducible. For example, ${x^2 + 1}$ cannot be factored using real numbers.

Advanced Techniques and Applications

Factoring by grouping is not just a standalone technique; it's a foundational skill that supports more advanced algebraic manipulations. Understanding this method can be beneficial in various areas of mathematics.

Applications in Solving Equations

Factoring is crucial in solving polynomial equations. Once a polynomial is factored, the zero-product property allows us to find the roots (solutions) of the equation. For example, if we have the equation:

${(x - 2)(x + 3) = 0}$

We can set each factor equal to zero:

${x - 2 = 0 \text{ or } x + 3 = 0}$

Solving these gives us the solutions ${x = 2}$ and ${x = -3}$.

Simplifying Rational Expressions

Factoring is also essential when simplifying rational expressions (fractions with polynomials in the numerator and/or denominator). By factoring both the numerator and the denominator, we can cancel out common factors, thus simplifying the expression. For instance, consider the expression:

${\frac{x^2 - 4}{x^2 + 4x + 4}}$

Factoring both polynomials, we get:

${\frac{(x - 2)(x + 2)}{(x + 2)(x + 2)}}$

We can cancel out the common factor ${(x + 2)}$, simplifying the expression to:

${\frac{x - 2}{x + 2}}$

Calculus and Beyond

In higher-level mathematics, such as calculus, factoring is frequently used in various contexts, including finding limits, derivatives, and integrals. A strong foundation in factoring techniques can make these advanced topics more accessible.

Conclusion

Factoring by grouping is a powerful technique for factoring polynomials, particularly those with four or more terms. Faelyn's attempt highlights the importance of careful grouping and identifying the greatest common factor. By understanding the steps involved and being mindful of common mistakes, one can master this technique. Remember, the key to successful factoring lies in practice and a solid grasp of algebraic principles. For the polynomial ${6x^4 - 8x^2 + 3x^2 - 4}$, the correct factored form is ${(3x^2 - 4)(2x^2 + 1)}$. Factoring not only simplifies algebraic expressions but also lays the groundwork for solving equations and tackling more complex mathematical problems.