Factoring $-3v^2 + 2v + 21$: A Step-by-Step Guide

Factoring quadratic expressions is a fundamental skill in algebra. It involves breaking down a quadratic expression into a product of simpler expressions, typically binomials. This guide will walk you through the process of completely factoring the quadratic expression $-3v^2 + 2v + 21$. Let's dive in and make this concept crystal clear.

Understanding the Basics of Factoring Quadratics

Before we tackle the specific expression, let's recap the basics. A quadratic expression generally takes the form of $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Factoring this expression means rewriting it as $(px + q)(rx + s)$, where $p$, $q$, $r$, and $s$ are also constants. The goal is to find these constants such that the product of the binomials equals the original quadratic expression.

In the given expression, $-3v^2 + 2v + 21$, we have $a = -3$, $b = 2$, and $c = 21$. The negative leading coefficient adds a bit of complexity, but don't worry; we'll address it methodically. Factoring is essentially the reverse process of expanding binomials using the distributive property (often remembered by the acronym FOIL – First, Outer, Inner, Last). To master factoring, it’s essential to practice and become comfortable with recognizing patterns and applying different techniques. Remember, the key is to break down the problem into smaller, manageable steps.

Factoring is not just an algebraic trick; it is a powerful tool used in solving quadratic equations, simplifying expressions, and understanding the behavior of polynomial functions. When we factor a quadratic expression and set it equal to zero, we can find the roots or x-intercepts of the corresponding quadratic function. These roots provide crucial information about the graph of the function, including where it crosses the x-axis. Moreover, factoring helps in simplifying complex algebraic expressions by canceling out common factors in the numerator and denominator. This simplification is particularly useful in calculus and other advanced mathematical fields. In addition, factoring is used in various real-world applications, such as modeling projectile motion, optimizing areas and volumes, and solving engineering problems. Understanding the principles of factoring unlocks a wide range of problem-solving capabilities, making it an indispensable skill for anyone pursuing mathematics, science, or engineering.

Step-by-Step Factoring of $-3v^2 + 2v + 21$

1. Factor out the Greatest Common Factor (GCF)

The first step in any factoring problem is to look for a greatest common factor (GCF) among all the terms. This simplifies the expression and makes subsequent steps easier. In our case, the terms are $-3v^2$, $2v$, and $21$. The coefficients are -3, 2, and 21. The greatest common factor of these numbers is 1. However, notice that the leading coefficient is negative, which is often a bit inconvenient for factoring. To make things easier, we can factor out a -1 from the entire expression:

$-3v^2 + 2v + 21 = -1(3v^2 - 2v - 21)$

Now we have a positive leading coefficient inside the parentheses, which will simplify the factoring process. Factoring out the GCF not only simplifies the expression but also ensures that we are working with smaller numbers, reducing the chances of making errors. This initial step is often overlooked, but it can significantly streamline the factoring process, especially with more complex quadratic expressions.

2. Factoring the Trinomial $3v^2 - 2v - 21$

Now we need to factor the trinomial inside the parentheses: $3v^2 - 2v - 21$. This trinomial has a leading coefficient (3) that is not 1, so we'll use the ac method.

a. Multiply a and c

Multiply the leading coefficient ($a = 3$) by the constant term ($c = -21$):

$3 imes -21 = -63$

This product, -63, is the key number we'll use to find the right factors. The ac method is particularly useful when the leading coefficient is not 1 because it provides a systematic way to find the correct factors that will allow us to rewrite the middle term. Understanding this method is crucial for factoring a wide range of quadratic expressions, especially those that don't fit the simpler patterns.

b. Find Two Numbers

We need to find two numbers that multiply to -63 and add up to the middle coefficient ($b = -2$). Let's list the factor pairs of -63:

• 1 and -63
• -1 and 63
• 3 and -21
• -3 and 21
• 7 and -9
• -7 and 9

Among these pairs, 7 and -9 add up to -2. These are the numbers we need! Finding the correct pair of numbers is the most critical step in the ac method. It may require some trial and error, but a systematic approach, such as listing factor pairs, helps to ensure that we don't miss the correct combination. Once we have the correct numbers, we can proceed to rewrite the middle term and complete the factoring process.

c. Rewrite the Middle Term

Rewrite the middle term (-2v) using the two numbers we found (7 and -9):

$3v^2 - 2v - 21 = 3v^2 - 9v + 7v - 21$

We've split the middle term into two terms, effectively turning the trinomial into a four-term polynomial. This rewriting is the core of the ac method and sets the stage for factoring by grouping. The choice of how to split the middle term is guided by the numbers we found in the previous step, ensuring that the subsequent factoring by grouping will work out smoothly.

d. Factor by Grouping

Now, we factor by grouping. Group the first two terms and the last two terms:

$(3v^2 - 9v) + (7v - 21)$

Factor out the GCF from each group:

$3v(v - 3) + 7(v - 3)$

Notice that both terms now have a common factor of $(v - 3)$. This is a crucial checkpoint in the factoring process. If the binomials inside the parentheses are not the same, it indicates an error in the previous steps, and we need to revisit our work. Seeing the common factor is a sign that we are on the right track.

e. Final Factorization

Factor out the common binomial factor $(v - 3)$:

$(3v + 7)(v - 3)$

So, $3v^2 - 2v - 21$ factors to $(3v + 7)(v - 3)$. This step completes the factorization of the trinomial. We have successfully expressed the trinomial as a product of two binomials. Double-checking the result by expanding the product can confirm the accuracy of our factorization.

3. Incorporate the -1

Remember, we factored out a -1 in the first step. We need to include it in our final answer:

$-1(3v^2 - 2v - 21) = -1(3v + 7)(v - 3)$

We can also distribute the -1 into one of the factors to get rid of the negative sign outside. Let's distribute it into the first factor:

$-(3v + 7)(v - 3) = (-3v - 7)(v - 3)$

Alternatively, we could distribute the -1 into the second factor:

$-(3v + 7)(v - 3) = (3v + 7)(-v + 3)$

Both $(-3v - 7)(v - 3)$ and $(3v + 7)(-v + 3)$ are equivalent factored forms of the expression. The choice of which form to use often depends on the context of the problem or personal preference. The key is to ensure that the final factored form is equivalent to the original expression.

Final Answer

Therefore, the completely factored form of $-3v^2 + 2v + 21$ is $(-3v - 7)(v - 3)$ or $(3v + 7)(-v + 3)$.

Factoring quadratic expressions can seem daunting at first, but with a systematic approach and practice, you can master it. Remember to always look for a GCF first, use the ac method when the leading coefficient is not 1, and double-check your work. Factoring is a critical skill in algebra, opening doors to solving equations, simplifying expressions, and understanding mathematical relationships.

Conclusion

In this guide, we've walked through the step-by-step process of completely factoring the quadratic expression $-3v^2 + 2v + 21$. We covered the importance of factoring out the GCF, using the ac method to handle leading coefficients, and the technique of factoring by grouping. By understanding these steps and practicing regularly, you can confidently tackle a wide range of factoring problems.

For further learning and practice, you might find helpful resources at websites like Khan Academy Algebra, which offers comprehensive lessons and exercises on factoring and other algebraic topics.