# Synthetic Division

How to use the synthetic division to divide a polynomial by a binomial: examples and their solutions.

## Example 1: Simplify (*x*^{3} - 7*x* + 11)/(*x* - 2)

Synthetic division is a way

to divide a polynomial by a binomial:

just like long division.

Let's solve this example,

which you've solved before,

by using synthetic division.

Long division

The given expression means

(1*x*^{3} + 0*x*^{2} - 7*x* + 11) ÷ (*x* - 2).

Write the coefficients of the numerator terms:

1 0 -7 11.

Don't forget to write the 0 from the *x*^{2} term.

Write the L shape form like this.

And write the zero of the divisor (*x* - 2):

2.

Write 1

below the form.

Write 1⋅2 = 2

below 0.

Write 0 + 2 = 2

below the form.

Write 2⋅2 = 4

below -7.

Write -7 + 4 = -3

below the form.

Write -3⋅2 = -6

below 11.

Write 11 - 6 = 5

below the form.

Draw another L shape form

that covers the 5.

[1 2 -3] means the coefficients of the quotient.

So the quotient is [1*x*^{2} + 2*x* - 3].

5 is the remainder.

And 2 is the zero of the divisor.

So write [+ 5/(*x* - 2)].

So (given) = *x*^{2} + 2*x* - 3 + 5/(*x* - 2).

## Example 2: Simplify (2*x*^{4} + *x*^{3} - 5*x*^{2} + 3*x* + 4)/(*x* + 1)

The given expression means

(2*x*^{4} + *x*^{3} - 5*x*^{2} + 3*x* + 4) ÷ (*x* - (-1)).

Write the coefficients of the dividend terms:

2 1 -5 3 4.

Write the L shape form like this.

And write the zero of the divisor (*x* - (-1)):

-1.

Write 2

below the form.

Write 2⋅(-1) = -2

below 1.

Write 1 - 2 = -1

below the form.

Write (-1)⋅(-1) = 1

below -5.

Write -5 + 1 = -4

below the form.

Write (-4)⋅(-1) = 4

below 3.

Write 3 + 4 = 7

below the form.

Write 7⋅(-1) = -7

below 4.

Write 4 - 7 = -3

below the form.

Draw another L shape form

that covers the -3.

[2 -1 -4 7] means the coefficients of the quotient.

So the quotient is [2*x*^{3} - 1*x*^{2} - 4*x* + 7].

-3 is the remainder.

And -1 is the zero of the divisor.

So write [+ (-3)/(*x* - (-1))].

-(-1) = +1

So (given) = 2*x*^{3} - *x*^{2} - 4*x* + 7 - 3/(*x* + 1).