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:

\({p(x)\over q(x)} = {Q(x) + {R\over q(x)}}\)
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.
 

The key differences between synthetic division and long division are summarized in the table below
 

Let’s see the step-by-step method for synthetic division using an example. 
Divide \((2x^3 + 5x^2 - 3x + 4) {\text { by }} (x - 1) \)

Step 1: Check the polynomial

Check whether the given polynomial is in standard form. All terms should be arranged in descending order of powers.

The given polynomial is \(2x^3 + 5x^2 - 3x + 4\). Now write down only the coefficients \(2x^3 + 5x^2 - 3x + 4 \) 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

\( \begin{array}{r|rrrr} & & & & \\ \hline & & & & \\ 1 & 2 & 5 & -3 & 4 \\ \end{array} \)

Step 3: Bring down the first number

Just bring down the first number 

\( \begin{array}{r|rrrr} & & & & \\ \hline & & & & \\ 1 & 2 & 5 & -3 & 4 \\ &\downarrow\\ & &2 \end{array} \)

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.

\( \begin{array}{r|rrrr} & & & & \\ \hline & & & & \\ 1 & 2 & 5 & -3 & 4 \\ &\downarrow &2 & 7 & 4 \\ & &2 & 7 & 4 & 8 \\ \end{array} \)

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

\(2x^2 + 7x + 4\)

The last number, 8, is the remainder.

The final answer should be in the format of:

\({p(x)\over q(x)} = {Q(x) + {R\over q(x)}} \)
\({{2x^3+5x^2-3x+4\over x-1}} = (2x^2 + 7x + 4) + {{8\over (x - 1)}} \)

Synthetic division simplifies dividing a polynomial by a linear binomial like \((x - 20)\). 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 \(x^2\) 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 \((x - 2)\). It cannot be used for complicated divisors like \((x^2 + x)\).

• 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.
   

Synthetic division is a quick and effect method to divide polynomials. In this section, we will learn some tips and tricks to make synthetic division. 

• Identify the value of r from the divisor \(x - r\). Remember to interchange the sign of the constant in the binomial to find r. For example, if the divisor is \(x - 3\) then r = 3. 
  
   
• Write the coefficients carefully, also include zeros for missing terms. For example, \(2x^3 + 0x^2 - 5x + 1\) becomes 2, 0, -5, 1. 
  
   
• Always remember that the last number in the row represents the remainder, while the other numbers give the coefficients of the quotient polynomials. Always double-check the answer using the formula: \(\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \).
  
   
• Start practice with small coefficient and positive divisors to understand the process. 
  
   
• After solving, verify the answer using long division method. If the results are same then the calculation is correct.   

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 the applications: 

• Healthcare: In MRI machines and ultrasound systems, signals are represented as polynomials functions. Synthetic division helps process and simplify these signals, removing noise and improving clarity. Synthetic division makes it easier, and the doctor gets clearer images and more accurate readings.

• Aerospace: Polynomials are used to calculate the flight path of rockets, aircraft, and drones. Synthetic division simplifies complex motion equations to make quicker decisions about rerouting and correcting the flight path. 

• Robotics: Robots use polynomial equations to model and control their movements. Synthetic division helps in analyzing these equations quickly, improving motion accuracy, stability, and error detection in robotic systems.

• Computer Graphics: Polynomials are used to describe curves and motion paths. Synthetic division simplifies these equations, helping designers create smoother animations, realistic simulations, and accurate 3D models in games and movies.

• Engineering: Engineers use synthetic division to test and modify polynomial models representing system performance, such as stress on bridges or load distribution in machines.