Factoring $42x^3 - 24x^2 - 35x + 20$: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of factoring polynomials, and we're going to tackle a specific expression: $42x^3 - 24x^2 - 35x + 20$. Factoring might seem daunting at first, but don't worry, we'll break it down step by step. By the end of this guide, you'll have a solid understanding of how to factor this expression and similar ones. Let's get started!

Understanding Factoring

Before we jump into the problem, let's quickly recap what factoring is all about. In simple terms, factoring is like reverse multiplication. Think of it as taking a polynomial and breaking it down into smaller expressions (factors) that, when multiplied together, give you the original polynomial. This is super useful in algebra for solving equations, simplifying expressions, and understanding the behavior of functions. When you factor polynomials, you're essentially trying to find those smaller expressions that multiply back to give you the original polynomial. There are several techniques to approach this, and the best method often depends on the specific polynomial you're working with. For our expression, $42x^3 - 24x^2 - 35x + 20$, we'll be using a technique called factoring by grouping. This involves grouping terms together and looking for common factors within those groups.

Method 1: Factoring by Grouping

Step 1: Group the Terms

The first step in factoring by grouping is, well, grouping! We're going to group the first two terms and the last two terms together. This is a common strategy when you have a polynomial with four terms. So, for our expression $42x^3 - 24x^2 - 35x + 20$, we'll group it like this:

$(42x^3 - 24x^2) + (-35x + 20)$

Notice how we've kept the signs consistent. The negative sign in front of the $35x$ stays with it inside the group. This is crucial for the next steps.

Step 2: Factor out the Greatest Common Factor (GCF) from Each Group

Now comes the fun part – finding the greatest common factor (GCF) for each group. The GCF is the largest factor that divides evenly into all terms in the group. Let's look at our first group, $(42x^3 - 24x^2)$.

• The GCF of 42 and 24 is 6.
• The GCF of $x^3$ and $x^2$ is $x^2$.

So, the GCF of the entire group is $6x^2$. We'll factor that out:

$6x^2(7x - 4)$

Now let's tackle the second group, $(-35x + 20)$.

• The GCF of -35 and 20 is -5 (we factor out a negative to make the next step easier).

Factoring out -5, we get:

$-5(7x - 4)$

Step 3: Notice the Common Binomial Factor

This is where the magic happens! Look closely at the expressions we have now:

$6x^2(7x - 4) - 5(7x - 4)$

Do you see it? The binomial $(7x - 4)$ is common to both terms. This is the key to factoring by grouping. If you don't have a common binomial factor at this stage, double-check your work – you might need to adjust your grouping or GCF.

Step 4: Factor out the Common Binomial Factor

Since $(7x - 4)$ is a common factor, we can factor it out just like we did with the GCF earlier. Think of $(7x - 4)$ as a single unit. Factoring it out, we get:

$(7x - 4)(6x^2 - 5)$

And there you have it! We've successfully factored the expression.

Step 5: Check Your Answer

It's always a good idea to check your work, especially in math. To check our factored form, we'll multiply the factors back together using the distributive property (or the FOIL method):

$(7x - 4)(6x^2 - 5) = 7x(6x^2) + 7x(-5) - 4(6x^2) - 4(-5)$

$= 42x^3 - 35x - 24x^2 + 20$

Rearranging the terms, we get:

$42x^3 - 24x^2 - 35x + 20$

This matches our original expression, so we know our factoring is correct!

Method 2: Alternative Grouping (If Applicable)

Sometimes, the initial grouping might not lead to a common binomial factor. In such cases, you can try rearranging the terms and grouping them differently. However, in this specific example, the original grouping works perfectly fine, so we don't need to explore alternative groupings. But it's a good trick to keep in your back pocket for other factoring problems!

Common Mistakes to Avoid

Factoring can be tricky, and it's easy to make mistakes. Here are a few common pitfalls to watch out for:

1. Incorrectly Identifying the GCF: Make sure you're finding the greatest common factor, not just a common factor. For example, if you have $12x^2 + 18x$, the GCF is $6x$, not just $2x$ or $3x$.
2. Forgetting to Factor out a Negative: When the leading coefficient of a group is negative, it's often helpful to factor out a negative GCF. This can make the binomial factors match up correctly.
3. Not Distributing the Negative Sign: When you have a negative sign outside a parenthesis, remember to distribute it to all terms inside the parenthesis. For example, $-(x - 2)$ is equal to $-x + 2$, not $-x - 2$.
4. Stopping Too Early: Always double-check if your factored expression can be factored further. Sometimes, you might need to apply factoring techniques multiple times to get the fully factored form. Also, do not forget to check your work.
5. Mixing Up Factoring and Solving: Factoring is an expression, while solving is an equation. When factoring, the goal is to rewrite the expression in a different form. When solving, the goal is to find the values of the variable that make the equation true.

Practice Problems

To really master factoring, practice is key! Here are a few problems similar to the one we just worked through. Try factoring them on your own, and then check your answers.

1. $15x^3 + 10x^2 - 9x - 6$
2. $8x^3 - 12x^2 - 10x + 15$
3. $6x^3 + 9x^2 + 4x + 6$

Conclusion

Factoring the expression $42x^3 - 24x^2 - 35x + 20$ might have seemed intimidating at first, but by using the method of factoring by grouping, we broke it down into manageable steps. Remember, the key is to group the terms, find the GCF of each group, identify the common binomial factor, and factor it out. Always double-check your work to make sure you haven't made any mistakes. With practice, you'll become a factoring pro in no time!

So, guys, keep practicing, and don't be afraid to tackle those tough factoring problems. You've got this!