Factoring Polynomials: A Step-by-Step Guide

Nov 12, 2025 by ADMIN 44 views

Hey guys! Factoring polynomials can seem daunting, but with a systematic approach, it becomes a breeze. Let's break down the polynomial $7x^4 + 14x^3 - 168x^2$ and find its completely factored form. We'll go through each step, so you can tackle similar problems with confidence. Ready? Let's dive in!

1. Identifying Common Factors

First things first, always look for common factors in all the terms of the polynomial. In our case, we have $7x^4$ , $14x^3$ , and $-168x^2$ . What’s common among them? We can see that each term is divisible by 7, and each term also contains at least $x^2$ . So, we can factor out $7x^2$ from the entire polynomial.

Factoring out $7x^2$ gives us:

$7x^2(x^2 + 2x - 24)$

Now, we have simplified the polynomial into a product of $7x^2$ and a quadratic expression $(x^2 + 2x - 24)$ . This is a great start! But we're not done yet. We need to factor the quadratic expression further to find the completely factored form. Trust me, it's like peeling an onion, layer by layer!

2. Factoring the Quadratic Expression

Next up, let's focus on factoring the quadratic expression $(x^2 + 2x - 24)$ . To factor a quadratic expression of the form $ax^2 + bx + c$ , we look for two numbers that multiply to $c$ (in this case, -24) and add up to $b$ (in this case, 2). This is where a little bit of mental math or a quick scan of possible factors comes in handy.

Think about the factors of -24. We need a pair of factors that have a difference of 2. How about 6 and -4? Let's check:

$6 \times -4 = -24$
$6 + (-4) = 2$

Bingo! These are the numbers we need. So, we can rewrite the quadratic expression as:

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

This means that $x^2 + 2x - 24$ can be factored into $(x + 6)(x - 4)$ . Awesome, right? We're getting closer to the completely factored form. Now, let's put it all together.

3. Combining the Factors

Now that we've factored both the common factor and the quadratic expression, we can combine them to get the completely factored form of the original polynomial. Remember, we factored out $7x^2$ in the first step, and we factored the quadratic expression into $(x + 6)(x - 4)$ .

So, the completely factored form of $7x^4 + 14x^3 - 168x^2$ is:

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

And that's it! We've successfully factored the polynomial completely. Take a moment to appreciate the simplicity and elegance of the factored form. It's like unlocking a secret code!

4. Matching with the Given Options

Alright, let's match our factored form with the given options to make sure we've nailed it. The options were:

A. $7x^3(x + 4)(x - 6)$
B. $7x^3(x - 4)(x + 6)$
C. $7x^2(x - 4)(x + 6)$
D. $7x^2(x + 4)(x - 6)$

Comparing our factored form $7x^2(x + 6)(x - 4)$ with the options, we can see that option C matches perfectly. Note that the order of the factors doesn't matter, since multiplication is commutative. So, $(x + 6)(x - 4)$ is the same as $(x - 4)(x + 6)$ .

Therefore, the correct answer is C. $7x^2(x - 4)(x + 6)$ .

5. Common Mistakes to Avoid

Factoring polynomials can be tricky, and it's easy to make mistakes if you're not careful. Here are a few common mistakes to watch out for:

Forgetting to Factor Out Common Factors: Always look for common factors first! This simplifies the polynomial and makes it easier to factor further. If you miss this step, you might end up with a more complicated expression to factor.
Incorrectly Factoring the Quadratic Expression: Make sure you find the correct pair of numbers that multiply to $c$ and add up to $b$ . A sign error or incorrect factors can lead to the wrong factored form. Double-check your work!
Not Factoring Completely: Ensure that you have factored the polynomial completely. Sometimes, after factoring once, you might still have a factorable expression. Keep factoring until you can't factor any further.
Sign Errors: Pay close attention to the signs of the terms. A simple sign error can throw off the entire factoring process. Be meticulous and double-check your signs.
Rushing Through the Process: Take your time and don't rush. Factoring requires attention to detail, and rushing can lead to mistakes. Work methodically and double-check each step.

By avoiding these common mistakes, you can improve your accuracy and confidence in factoring polynomials. Remember, practice makes perfect! So, keep practicing, and you'll become a factoring pro in no time.

6. Practice Problems

Want to put your factoring skills to the test? Here are a few practice problems for you to try:

$3x^3 + 12x^2 - 15x$
$2x^4 - 8x^3 - 10x^2$
$5x^4 + 20x^3 + 15x^2$

Try factoring these polynomials completely, and then check your answers with a friend or teacher. The more you practice, the better you'll become at factoring. Remember, every problem is an opportunity to learn and improve. Keep pushing yourself, and you'll master factoring in no time!

7. Conclusion

So, in conclusion, the completely factored form of the polynomial $7x^4 + 14x^3 - 168x^2$ is $7x^2(x - 4)(x + 6)$ . We found this by first identifying and factoring out the common factor $7x^2$ , and then factoring the resulting quadratic expression $(x^2 + 2x - 24)$ into $(x - 4)(x + 6)$ . Remember to always look for common factors first, and be careful when factoring quadratic expressions. Avoid common mistakes like sign errors and not factoring completely. And most importantly, practice, practice, practice!

With these steps and tips, you'll be well on your way to mastering factoring polynomials. Keep up the great work, and happy factoring, guys! You've got this!