Factoring Completely $-4x^2 + 16x - 24$ A Step-by-Step Guide

Factoring polynomials completely is a crucial skill in algebra. It simplifies expressions, solves equations, and helps analyze functions. In this article, we will walk through the process of completely factoring the quadratic expression $-4x^2 + 16x - 24$. We will explore the step-by-step method to identify common factors and factor the resulting quadratic expression. Understanding this process is fundamental for more advanced algebraic manipulations. Let's dive into the world of factoring!

Understanding the Problem

The problem at hand is to factor the quadratic expression $-4x^2 + 16x - 24$ completely. This means we need to break down the expression into its simplest multiplicative components. Factoring is the reverse process of expansion, where we express a polynomial as a product of two or more simpler polynomials or monomials. To factor completely, we must ensure that each factor cannot be factored further. This involves identifying the greatest common factor (GCF) and then factoring the remaining quadratic expression, if possible. Mastering the art of factoring is essential for simplifying algebraic expressions, solving quadratic equations, and understanding polynomial functions.

Step-by-Step Solution

1. Identify the Greatest Common Factor (GCF)

Our initial focus should be on identifying the greatest common factor (GCF) among all the terms in the expression $-4x^2 + 16x - 24$. The GCF is the largest factor that divides each term without leaving a remainder. Let's break down each term:

• $-4x^2$: Factors include -4, $x^2$
• $16x$: Factors include 16, $x$
• $-24$: Factors include -24

By examining the coefficients (-4, 16, and -24), we can determine that the greatest common numerical factor is -4. It's often beneficial to factor out a negative number if the leading coefficient is negative. This simplifies the expression and makes subsequent factoring steps easier. Therefore, the GCF for this expression is -4.

2. Factor Out the GCF

Now that we have identified the GCF as -4, the next step is to factor it out from the entire expression. This involves dividing each term of the expression by -4 and writing the expression as a product of -4 and the resulting polynomial:

$-4x^2 + 16x - 24 = -4(x^2 - 4x + 6)$

To verify that we have factored correctly, we can distribute the -4 back into the parentheses:

$-4(x^2 - 4x + 6) = -4 * x^2 + (-4) * (-4x) + (-4) * 6 = -4x^2 + 16x - 24$

This confirms that our factoring is accurate so far. Now, we need to examine the expression inside the parentheses, $x^2 - 4x + 6$, to see if it can be factored further.

3. Factor the Quadratic Expression (if possible)

After factoring out the GCF, we are left with the quadratic expression $x^2 - 4x + 6$. To determine if this quadratic can be factored further, we look for two numbers that multiply to the constant term (6) and add up to the coefficient of the linear term (-4). Let's list the factor pairs of 6:

• 1 and 6
• 2 and 3
• -1 and -6
• -2 and -3

Now, we check if any of these pairs add up to -4:

• 1 + 6 = 7
• 2 + 3 = 5
• -1 + (-6) = -7
• -2 + (-3) = -5

None of these pairs add up to -4. This indicates that the quadratic expression $x^2 - 4x + 6$ cannot be factored further using integer coefficients. It is a prime quadratic.

4. Final Factored Form

Since the quadratic expression $x^2 - 4x + 6$ cannot be factored further, the completely factored form of the original expression $-4x^2 + 16x - 24$ is:

$-4(x^2 - 4x + 6)$

This is the final answer. We have successfully factored out the GCF and determined that the remaining quadratic expression is irreducible over the integers.

Selecting the Correct Option

Based on our step-by-step solution, the completely factored form of $-4x^2 + 16x - 24$ is $-4(x^2 - 4x + 6)$. Now, let's compare this result with the given options:

A. $-1(x^2 - 16x + 24)$ B. $-4(x^2 - 4x + 6)$ C. $-4x(x^2 - 4x + 6)$ D. $-4(x^2 + 4x - 6)$

By comparing our solution with the provided options, we can clearly see that option B, $-4(x^2 - 4x + 6)$, matches our factored expression exactly. Therefore, option B is the correct answer.

Conclusion

In this article, we successfully factored the quadratic expression $-4x^2 + 16x - 24$ completely. We started by identifying the greatest common factor (GCF), which was -4. Factoring out the GCF simplified the expression to $-4(x^2 - 4x + 6)$. We then attempted to factor the quadratic expression $x^2 - 4x + 6$ further, but it turned out to be irreducible over the integers. Therefore, the final factored form is $-4(x^2 - 4x + 6)$.

The key takeaways from this exercise include:

1. Always look for the GCF first: Factoring out the GCF simplifies the expression and makes subsequent factoring easier.
2. Check if the resulting quadratic can be factored further: After factoring out the GCF, examine the remaining quadratic expression for further factoring possibilities.
3. Understand prime quadratics: Some quadratic expressions cannot be factored using integer coefficients, and these are called prime quadratics.

By following these steps and understanding the underlying principles of factoring, you can confidently tackle similar problems and enhance your algebraic skills. Factoring is a fundamental skill that paves the way for more advanced topics in mathematics, such as solving equations, simplifying rational expressions, and analyzing polynomial functions. Keep practicing, and you'll master the art of factoring completely!