Synthetic Division

Synthetic Division

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

Example 1: Simplify (x3 - 7x + 11)/(x - 2)

Simplify the given expression. (x^3 - 7x + 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
(1x3 + 0x2 - 7x + 11) ÷ (x - 2).

Write the coefficients of the numerator terms:
1 0 -7 11.

Don't forget to write the 0 from the x2 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 [1x2 + 2x - 3].

5 is the remainder.
And 2 is the zero of the divisor.
So write [+ 5/(x - 2)].

So (given) = x2 + 2x - 3 + 5/(x - 2).

Example 2: Simplify (2x4 + x3 - 5x2 + 3x + 4)/(x + 1)

Simplify the given expression. (2x^4 + x^3 - 5x^2 + 3x + 4)/(x + 1)

The given expression means
(2x4 + x3 - 5x2 + 3x + 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 [2x3 - 1x2 - 4x + 7].

-3 is the remainder.
And -1 is the zero of the divisor.
So write [+ (-3)/(x - (-1))].

-(-1) = +1

So (given) = 2x3 - x2 - 4x + 7 - 3/(x + 1).