# Synthetic Division

## Related Calculators: Polynomial Calculator , Synthetic Division Calculator

Synthetic Division is a method, similar to polynomial long division, but it requires less writing and fewer calculations. However, it can be used only for dividing polynomial in one variable by linear polynomial x-a.

• less space on a paper
• fewer calculations
• calculations are made without variables

Disadvantage: it is only used for dividing polynomial by LINEAR polynomial. In other cases it won't work.

Example 1. Divide x^2-7x+10 by x-2, using synthetic division.

First, we need to order degrees of polynomial from greatest to lowest.

It is already done: color(green)(1)x^2color(cyan)(-7)xcolor(purple)(+10).

Also, color(brown)(a=2).

Next, write coefficients in the special form:

$$\begin{array}{r|lll}\phantom{2}&x^2&x^1&x^0\\\color{brown}{2}&\color{green}{1}&\color{cyan}{-7}&\color{purple}{10}\\\phantom{1}&\phantom{-7}&\phantom{10}\\\hline \end{array}$$$Now, perform the following steps:  Take coefficient of the leading term and carry it down unchanged. $$\begin{array}{r|lll}2&1&-7&10\\\phantom{2}&\color{red}{\downarrow}&\phantom{-7}&\phantom{10}\\\hline\phantom{2}&\color{red}{1}&\phantom{-7}&\phantom{10}\end{array}$$$ Multiply carry-down value by a: 1*2=2, and write result into the next column. $$\begin{array}{r|lll}\color{red}{2}&1&-7&10\\\phantom{2}&\downarrow&\color{red}{+2}&\phantom{10}\\\hline\phantom{2}&\color{red}{1^{\nearrow}}&\phantom{-7}&\phantom{10}\end{array}$$$Add down the column. $$\begin{array}{r|lll}2&1&\color{red}{-7}&10\\\phantom{2}&\downarrow&\color{red}{+2}&\phantom{10}\\\hline \phantom{2}&1^{\nearrow}&\color{red}{-5}&\phantom{10}\end{array}$$$ Multiply calculated result by a: (-5)*2=-10, and write the result into the next column. $$\begin{array}{r|lll}\color{red}{2}&1&-7&\phantom{-}10\\\phantom{2}&\downarrow&+2&\color{red}{-10}\\\hline \phantom{2}&1^{\nearrow}&\color{red}{-5^{\nearrow}}&\phantom{10}\end{array}$$$Add down the column. $$\begin{array}{r|lll}2&1&-7&\phantom{-}\color{red}{10}\\\phantom{2}&\downarrow&+2&\color{red}{-10}\\\hline \phantom{2}&1^{\nearrow}&-5^{\nearrow}&\phantom{-1}\color{red}{0}\end{array}$$$

We are done.

Coefficients under the horizontal line (except last) are coefficients of the quotient. Last coefficient is a remainder (it is zero).

Thus, quotient is 1x-5 or simply x-5.

Answer: (x^2-7x+10)/(x-2)=x-5.

Sometimes, there will be missing terms.

For example, in x^3-1 missing terms are the terms, that involve x^2 and x.

To fix that, just add missing terms with zero coefficient. This doesn't change anything.

Example 2. Divide (-5x+3x^3-4)/(x+1).

Order degrees of dividend from greatest to lowest: 3x^3-5x-4.

There is missing term, involving x^2, so we add it with zero coefficient: 3x^3+color(red)(0x^2)-5x-4.

Since, x+1=x-(-1), then a=-1.

Next, write coefficients in the special form:

$$\begin{array}{r|llll}\phantom{-1}&x^3&x^2&x^1&x^0\\-1&3&0&-5&-4\\\phantom{1}&\phantom{-7}&\phantom{10}\\\hline \end{array}$$$Now, perform the following steps:  Take coefficient of the leading term and carry it down unchanged. $$\begin{array}{r|llll}-1&3&0&-5&-4\\\phantom{-1}&\color{red}{\downarrow}&\phantom{0}&\phantom{-5}&\phantom{-4}\\\hline\phantom{-1}&\color{red}{3}&\phantom{0}&\phantom{-5}&\phantom{-4}\end{array}$$$ Multiply carry-down value by a: 3*(-1)=-3, and write result into the next column. $$\begin{array}{r|llll}\color{red}{-1}&3&\phantom{-}0&-5&-4\\\phantom{-1}&\downarrow&\color{red}{-3}&\phantom{-5}&\phantom{-4}\\\hline\phantom{-1}&\color{red}{3^{\nearrow}}&\phantom{-3}&\phantom{-5}&\phantom{-4}\end{array}$$$Add down the column. $$\begin{array}{r|llll}-1&3&\phantom{-}\color{red}{0}&-5&-4\\\phantom{-1}&\downarrow&\color{red}{-3}&\phantom{-5}&\phantom{-4}\\\hline\phantom{-1}&3^{\nearrow}&\color{red}{-3}&\phantom{-5}&\phantom{-4}\end{array}$$$ Multiply calculated result by a: (-3)*(-1)=3, and write the result into the next column. $$\begin{array}{r|llll}\color{red}{-1}&3&\phantom{-}0&-5&-4\\\phantom{-1}&\downarrow&-3&\color{red}{+3}&\phantom{-4}\\\hline\phantom{-1}&3^{\nearrow}&\color{red}{-3^{\nearrow}}&\phantom{-5}&\phantom{-4}\end{array}$$$Add down the column. $$\begin{array}{r|llll}-1&3&\phantom{-}0&\color{red}{-5}&-4\\\phantom{-1}&\downarrow&-3&\color{red}{+3}&\phantom{-4}\\\hline\phantom{-1}&3^{\nearrow}&-3^{\nearrow}&\color{red}{-2}&\phantom{-4}\end{array}$$$ Multiply calculated result by a: (-2)*(-1)=2, and write the result into the next column. $$\begin{array}{r|llll}\color{red}{-1}&3&\phantom{-}0&-5&-4\\\phantom{-1}&\downarrow&-3&+3&\color{red}{+2}\\\hline\phantom{-1}&3^{\nearrow}&-3^{\nearrow}&\color{red}{-2^{\nearrow}}&\phantom{-4}\end{array}$$$Add down the column. $$\begin{array}{r|llll}-1&3&\phantom{-}0&-5&\color{red}{-4}\\\phantom{-1}&\downarrow&-3&+3&\color{red}{+2}\\\hline\phantom{-1}&3^{\nearrow}&-3^{\nearrow}&-2^{\nearrow}&\color{red}{-2}\end{array}$$$

We are done.

Coefficients under the horizontal line (except last) are coefficients of the quotient. Last coefficient is a remainder (it is non-zero).

Thus, quotient is 3x^2-3x-2 and remainder is -2.

Therefore, we can write, that (3x^3-5x-4)=(3x^2-3x-2)(x+1)-2.

Or (3x^3-5x-4)/(x+1)=3x^2-3x-2-2/(x+1).

Answer: (3x^3-5x-4)/(x+1)=3x^2-3x-2-2/(x+1).

Last example will be solved faster.

Example 3. Find quotient and remainder of (-10+4x+x^4-3x^3)/(x-3).

Order degrees of polynomial from greatest to lowest: x^4-3x^3+4x-10.

There is missing term, involving x^2, so we add it with zero coefficient: x^4-3x^3+color(red)(0x^2)+4x-10.

Also, a=3.

Table for synthetic division is the following:

$$\begin{array}{r|lllll}3&1&-3&\phantom{-}0&\phantom{-}4&-10\\\phantom{-1}&\downarrow&+3&+0&+0&+12\\\hline\phantom{3}&1^{\nearrow}&\phantom{-}0^{\nearrow}&\phantom{-}0^{\nearrow}&\phantom{-}4^{\nearrow}&\phantom{+1}2\end{array}$$\$

So, quotient is 1x^3+0x^2+0x+4=x^3+4 and remainder is 2.

Answer: (x^4-3x^2+4x-10)/(x-3)=x^3+4+2/(x-3).

Now, it is time to exercise.

Exercise 1. Use synthetic division to divide x^2-2x+5 by x-3.

Answer: (x^2-2x+5)/(x-3)=x+1+8/(x-3).

Exercise 2. Divide the following: (x^3+8)/(x+2).

Answer: x^2-2x+4.

Exercise 3. Find quotient and remainder of (2x^4+7x^3-15x^2+x+9)/(x+5).

Answer: Quotient is 2x^3-3x^2+1 and remainder is 4.