103 LearnersLast updated on July 9th, 2025
Synthetic Division
Dividing polynomials quickly and easily is done using synthetic division. Normally, we use the long division method to divide polynomials, but it is lengthy and difficult to follow. Synthetic division is a shortcut. This article discusses synthetic division and the steps involved.
What is the Synthetic Division?
We use synthetic division instead of the long division method because it is faster, and we don’t have to write each variable and its exponents repeatedly. Since we only work with coefficients in synthetic division, the process becomes easier. When we divide one polynomial by another, we can write it as:
p(x)/q(x) = Q(x) + R/q(x)
Here, p(x) is the dividend
q(x) is the divisor
Q(x) is the quotient we get from the division.
R is the remainder.
Difference Between Synthetic Division and Long Division
The key differences between synthetic division and long division are summarized in the table below
|
Aspect |
Synthetic Division |
Long Division |
|
What’s it for |
Synthetic division is the fastest way to divide when the divisor looks like (x-number). |
Long division is used to divide any kind of polynomial. |
|
What you need |
The divisor must be simple, like (x - 2). |
Long division can be any polynomial, like (x2 + x + 2). |
|
How it works |
Uses only the numbers, not the variables. |
You have to manage variables and their powers during division. |
|
Speed |
It is much faster for simple divisors. |
It takes more time and writing. |
|
Number of steps |
It involves fewer steps and relies primarily on addition and multiplication. |
More steps, divide, multiply, subtract, and bring the terms down. |
|
Best for |
When dividing by (x - number). |
Best for any type of polynomial. |
Explain Synthetic Division Step by Step
Let’s see the step-by-step method for synthetic division using an example.
Divide (2x3 + 5x2 - 3x + 4) by (x - 1)
Step 1: Check the polynomial
Check whether the given polynomial is in the standard form, that is, all the terms are arranged in descending order of powers. The given polynomial is 2x3 + 5x2 - 3x + 4. Now write down only the coefficients
2x3 + 5x2 - 3x + 4 becomes 2, 5, -3, 4
And the given divisor is (x - 1), we solve x - 1= 0 and get x = 1. This makes dividing easier.
Step 2: Set the synthetic division box
We have to create a division box with the divisor and the dividend.
Divisor: 1
Dividend: 2, 5, -3, 4
Step 3: Bring down the first number
Just bring down the first number
Step 4: Multiply and add
Multiply the first number that we brought down by the divisor and write the answer under the next number.
1 × 2 = 2
Now we have to write the 2 below the next number that is 5, and add both numbers together.
5 + 2 = 7
Now do the same multiplication and addition with the number 7
1 × 7 = 7
-3 + 7 = 4
And again, repeat the steps with 4
1 × 4 = 4
4 + 4 = 8
We get the number 8, which is the remainder, and the first three numbers will be the quotient.
Step 5: Write the final answer
The numbers at the bottom, except the last number, are quotient coefficients. We started the polynomial with a degree of 3, now we have to go one power lower. Therefore, the quotient becomes
2x2 + 7x + 4
The last number 8 is the remainder.
The final answer should be in the format of:
p(x)/q(x) = Q(x) + R/q(x)
2x3+5x2-3x+4/x-1 = (2x2 + 7x + 4) + 8/(x - 1)
What are the Advantages and Disadvantages of Synthetic Division?
If you have to divide a polynomial by a linear binomial, in the form of (x - 20), then synthetic division can help us get the answer much more easily and quickly. Here are some advantages and disadvantages of synthetic division.
Advantages of Synthetic Division
- Easier and Faster: Synthetic division is a quicker and easier way of dividing polynomials.
- Simple Steps: In synthetic division, we don't have to write all the variables like x2 or x. We just write the coefficients.
- Less Confusing: It is easier to divide with synthetic division because it has easier steps to follow and makes fewer mistakes, especially when we are careful with numbers.
- Useful for Certain Problems: It works great when dividing by something simple like (x - 2), and it can help with equations or simplify tricky expressions.
Disadvantages of Synthetic Division
- Works in some cases: We can use synthetic division only when the divisor is in the form of (x - 2), we can’t use the synthetic division method for complicated divisors like (x2 + x).
- Need to Know the Basics: We have to understand how polynomials work and how to write just the coefficients.
- Possibilities for Making Mistakes: If we write down the wrong number, the whole thing can go wrong, so while writing the coefficients, we have to be precise with the number and the signs.
Real Life Applications for Synthetic Division
Synthetic division is not only used in math, but it also plays a major role in several real-life applications, especially in fields like engineering, healthcare, etc. Here are some of them
- Healthcare: In MRI machines and ultrasound systems, signals are captured and processed using polynomials. To clean up the signals, polynomial signals are used. Synthetic division makes it easier, and the doctor gets clearer images and more accurate readings.
- Aerospace: Polynomials are used to calculate the path of rockets or drones. Synthetic division helps in simplifying complex motion equations to make quicker decisions about rerouting and correcting the flight path.
- Robotics: Polynomials are used to model and control robotic movements. To improve the control and detect errors, these polynomials can be analyzed and adjusted.
Common Mistakes and How to Avoid Them in Synthetic Division
Kids often make mistakes when doing synthetic division. Here are some of those mistakes and ways to avoid them, which can help them fix and understand them.
Mistake 1
Using the wrong sign for the divisor
While changing the divisor from (x - 3), the value of x, the divisor will become x = 3, but instead of writing 3, students might write it as -3 without changing the sign. Use only the number for the divisor; x-3 becomes 3, and x + 3 becomes -3.
Mistake 2
Mixing the order
While writing the coefficient, we have to write the polynomial in the standard form and take the coefficient. Children will sometimes change the order or don't keep the polynomial in the standard form and write down the coefficient, which will lead to a mistake. Always write the polynomial in descending order.
Mistake 3
Missing zero terms
When the polynomial is like x3 + 2x + 3, the x2 term is missing, so in that place we should add 0. Sometimes kids forget to add that 0, and the whole division will be a mistake.
Mistake 4
Doing subtraction instead of addition
After multiplying the number, students mistakenly subtract instead of adding. Synthetic division uses the multiply, add, multiply format.
Mistake 5
Dividing by higher-degree divisors
Synthetic division can only be done when the divisor is in the form of (x - c), where c is the number. It cannot be done with higher-degree polynomials like (x2 - x).
Solved Examples of Synthetic Division
Problem 1
Use synthetic division to divide: f(x) = 2x3 + 3x2 - 2x + 4 by x - 1.
Quotient: 2x2 + 5x + 3
Remainder: 7
Explanation
1. The coefficients are 2, 3, -2, 4.
2. The divisor is x - 1, i.e. x = 1.
3. Using the synthetic division method, divide them:
4. Write the quotient and the remainder. When writing the quotient, we have to reduce it to one power, and the last number is the remainder.
Quotient: 2x2 + 5x + 3
Remainder: 7
Problem 2
Use synthetic division to divide f(x) = 3x3 - 5x + 2 by x + 2.
Quotient: 3x2 -6x + 7
Remainder: -12
Explanation
1. Write the coefficients 3, 0, -5, 2. Here, the x2 term is missing, so we add 0 in its stead.
2. The given divisor is x + 2, so the divisor becomes -2.
Quotient: 3x2 -6x + 7
Remainder: -12
Problem 3
Divide 5x2 - 3x + 7 by x + 1.
Quotient: 5x - 8
Remainder: 15
Explanation
1. The coefficients are 5, -3, 7.
2. Dividing x + 1, so the divisor will become -1.
Quotient: 5x - 8
Remainder: 15
Problem 4
Divide 2x3 + 7x2 - x + 5 by x - 3
Quotient: 2x2 + 13x + 38
Remainder: 119
Explanation
1. The coefficients are 2, 7, -1, 5.
2. The divisor is 3.
Quotient: 2x2 + 13x + 38
Remainder: 119
Problem 5
Divide x2 + 2x + 1 by x + 1
Quotient: x + 1
Remainder: 0
Explanation
1. Coefficients: 1, 2, 1.
2. Divisor: -1
Quotient: x + 1
Remainder: 0
FAQs on Synthetic Division
1.What is synthetic division?
2.What are coefficients?
3.What is the last number in the synthetic division?
4.Why is it called synthetic division?
5.Why is synthetic division useful?
6.How does learning Algebra help students in Philippines make better decisions in daily life?
7.How can cultural or local activities in Philippines support learning Algebra topics such as Synthetic Division ?
8.How do technology and digital tools in Philippines support learning Algebra and Synthetic Division ?
9.Does learning Algebra support future career opportunities for students in Philippines?
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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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: He loves to play the quiz with kids through algebra to make kids love it.
