Synthetic Division: A Step-by-Step Guide

Oct 23, 2025 by SLV Team 41 views

Hey guys! Let's dive into the world of polynomial division, specifically using a super handy technique called synthetic division. We're going to break down how to divide the polynomial $6x^3 - 10x^2 + 20$ by the binomial $x + 1$ . Don't worry, it's not as scary as it sounds! Synthetic division is actually a streamlined way to do polynomial division, especially when you're dividing by a linear factor like our $x + 1$ . This method simplifies the process and gets you to the answer quickly. Plus, it's a fundamental concept in algebra, so understanding it will help you a lot in the long run. We'll go through each step carefully, so you'll feel confident using synthetic division in no time. Ready? Let's get started!

Setting Up the Synthetic Division Problem

Alright, first things first, let's get our problem set up correctly. The goal is to divide $6x^3 - 10x^2 + 20$ by $x + 1$ . In synthetic division, we use the zero of the divisor (the thing we're dividing by), which is $x + 1$ in this case. To find the zero, you set the divisor equal to zero and solve for x. So, $x + 1 = 0$ means $x = -1$ . We will use this -1 in our calculations. Now, we set up the division like this:

Write down the coefficients of the dividend (the polynomial we're dividing) in a row. Make sure to include all terms, even if they have a coefficient of zero. Our dividend is $6x^3 - 10x^2 + 20$ . Notice that there's no $x$ term. So, we need to include a 0 for the $x$ term. Our coefficients are 6, -10, 0, and 20. Arrange them like this: 6 -10 0 20
To the left of these coefficients, write the zero of the divisor (-1 in our case). This gives us: -1 | 6 -10 0 20

That's all for the setup. It might seem like a lot, but this is the hardest part. The rest is just simple arithmetic.

Performing the Synthetic Division

Now, let's get down to the actual division. Follow these steps carefully:

Bring down the first coefficient: Bring down the first coefficient (6 in our example) below the line. So we have:
```
-1 | 6  -10   0   20
    ---------
      6
```
Multiply and add: Multiply the number you just brought down (6) by the zero of the divisor (-1). Then, write the result (-6) under the next coefficient (-10). Add the numbers in that column (-10 and -6) and write the sum (-16) below the line. This looks like:
```
-1 | 6  -10   0   20
    -6   16  -16
    ---------
      6  -16
```
Repeat: Repeat the multiplication and addition process. Multiply -16 by -1 (which gives you 16), and write the result under the next coefficient (0). Add 0 and 16 to get 16. The work looks like:
```
-1 | 6  -10   0   20
    -6   16  -16
    ---------
      6  -16  16
```
One last time: Multiply 16 by -1 (which gives you -16), and write the result under the last coefficient (20). Add 20 and -16 to get 4. The full calculation looks like this:
```
-1 | 6  -10   0   20
    -6   16  -16
    ---------
      6  -16  16   4
```

Interpreting the Results

Great job! You've successfully performed the synthetic division. Now, let's figure out what those numbers mean.

The numbers below the line represent the coefficients of the quotient (the result of the division) and the remainder. The last number (4 in our case) is the remainder. The other numbers are the coefficients of the quotient, starting with a degree one less than the original polynomial. So, since we started with a cubic ( $x^3$ ), our quotient will start with an $x^2$ term. Therefore, the quotient is $6x^2 - 16x + 16$ , and the remainder is 4.

To write the final answer, we express the remainder as a fraction with the divisor ( $x + 1$ ) as the denominator. This gives us:

$6x^2 - 16x + 16 + rac{4}{x + 1}$

So, the correct answer from the choices provided is A. $6x^2 - 16x + 16 + rac{4}{x+1}$

That's it, guys! You've mastered synthetic division! Pretty cool, right? With a little practice, you'll be able to breeze through these problems.

Further Practice and Tips

Want to get even better? Here are a few tips and some practice problems to sharpen your skills:

Practice, Practice, Practice: The more problems you solve, the easier synthetic division becomes. Work through examples in your textbook or online resources.
Pay Attention to Signs: Be super careful with negative signs, especially when finding the zero of the divisor and during the multiplication steps.
Missing Terms: Always remember to include placeholders (with a coefficient of 0) for any missing terms in the dividend.
Check Your Work: After you find the answer, multiply the quotient by the divisor and add the remainder. This should give you the original dividend. This is a great way to check if your answer is correct.

Practice Problems:

Divide $x^3 - 7x^2 + 10x - 6$ by $x - 3$ .
Divide $2x^3 + 5x^2 - 8x - 1$ by $x + 4$ .

Give these problems a shot and see how you do. The more you practice, the more comfortable you'll become with synthetic division.

Why Synthetic Division Matters

Okay, so why is synthetic division such a big deal? Well, it's a super useful tool for a few reasons:

Efficiency: As we've seen, it's a much quicker method than long division, especially when dividing by a linear factor.
Finding Zeros: Synthetic division can also help you find the zeros (or roots) of a polynomial. If the remainder is zero, then the divisor is a factor of the polynomial, and the zero of the divisor is a zero of the polynomial.
Factoring Polynomials: Synthetic division can assist in factoring higher-degree polynomials. By identifying factors, you can break down complex expressions into simpler forms.
Applications: Synthetic division has applications in various fields, like engineering, computer science, and economics, where you might need to model and analyze polynomial functions.

So, learning synthetic division isn't just about passing a math test; it's about gaining a valuable skill that can be applied in many areas. Keep practicing, and you'll find it incredibly helpful.

Conclusion: You Got This!

Alright, we've covered a lot of ground today. We started with the basics of setting up synthetic division, then walked through the steps of performing the division, and finally, we interpreted the results to get the quotient and the remainder. Remember to take it one step at a time, and don't be afraid to pause and review the steps if you get stuck.

Synthetic division is a powerful tool, and with consistent practice, you'll become confident in using it to solve various polynomial division problems. Now go out there, tackle those practice problems, and show off your new skills! Keep up the great work, and keep exploring the amazing world of mathematics! You've got this, guys!