Synthetic Division for Polynomials

(Video) Synthetic Division

by PatrickJMT

This video by patrickJMT walks you through synthetic division using polynomials.

Synthetic division is a useful shortcut for long division. It emulates the process of long division by removing the variable terms from the polynomial. As a result you no longer have to distribute the constant multiple across the divisor.

(Video) Synthetic Division vs. Long Division

by Learn Math Tutorials

When dividing polynomials with more than one term in the divisor, it is often necessary to use either the method of Long Division or Synthetic Division. The video below from the YouTube channel Learn Math Tutorials explains the difference between these two methods of polynomial division and how to do the computations.

We have already learned multiple methods to divide polynomials, such as polynomial long division and polynomial division by factoring. Now, let's learn all about synthetic division!

##Synthetic Division

Synthetic division is another method of polynomial division. It is often simpler than polynomial long division and is used when the divisor is a polynomial of degree $1$.

##How It Works

Let's walk through an example.

We are given the expression $(3x^4 - 6x^3 + x^2 + 4x) \div (x-2)$.

As the divisor has a degree of $1$, we can do synthetic division.

To set up synthetic division, we set up what looks like a little house.

Let's input our example polynomial into this setup.

We know the root of $x-2$ is $2$ using the factor theorem for polynomials. Now that we have the setup, we can work on solving this problem.

To begin, bring the leading coefficient down below the line. Then, multiply the value below the line by the divisor, in this example $2$.

After we multiply, we can write the new value below the second coefficient, add the second coefficient to this new value, then write the result below the line. Then, repeat the pattern with the other coefficients.

We can use the numbers on the bottom row to get the quotient. Keep in mind that the last number is the remainder. We also know that the degree of the first coefficient of the quotient is one less than the highest degree of the dividend.

In our example, the quotient without the remainder would be $3x^3 + x +6$. Using the remainder theorem for polynomials, we can write the quotient and remainder together as:

$$3x^3 + x +6 +\frac{12}{x-2}$$

Synthetic Division for Polynomials

In this lesson, we are going to learn about synthetic division, a useful shortcut for polynomial division. It only works in certain cases, though, but it is much faster than polynomial long division.

We use synthetic division when we're dividing a polynomial by something like $x – c$, where $c$ is a number. For example, let's say we want to divide $2x³ + 3x² – 5x + 6$ by $x – 2$.

First, we write down the coefficients: $2, 3, -5, 6$. Then, on the left, we write down $2$, which is the opposite of $-2$. $2$ will be the number we divide by.

Next, you'll bring down the first number, $2$. Multiply $2$ by $2$ and write the number under $3$. Add the numbers in the second column, $3+4$, and write the sum under the line.

Keep doing this until you reach the last number. The last sum is the remainder of the division. The other numbers are the coefficients of your answer.

Synthetic division seems confusing at first, but it gets easier the more and more that you do it. It's a great tool if you want to check if something is a factor of a polynomial.