Polynomial long division

What is polynomial long division? Long division of polynomials is basically the same as numerical division. You follow the same long division algorithm consisting of the steps,"Divide, multiply, subtract, bring down, and repeat as necessary" in order to find a quotient and a remainder. 

Learn long division of polynomials here with examples and steps that are easy to follow and straight to the point. Take a look at the example below showing how to divide a second degree polynomial by a first degree polynomial. 

Explaining in more details the polynomial long division in the figure above. 

The polynomial long division in the figure above is a summary showing how to divide a trinomial by a binomial using long division. Read example #1 carefully so you can clearly see all the steps for long division of polynomials. In the end, you will fully understand it!

Example #1

Divide x² + 3x - 10 by x - 2

The leading term of the dividend (x² + 3x - 10) is x² and the leading term of the divisor (x - 2) is x

x² ÷ x = x

Write x as the first term of the quotient 

            x                         
 x - 2)  x² + 3x - 10

x(x - 2) = x² - 2x

Subtract x² - 2x from the dividend making sure that you line up the similar terms.

            x                         
 x - 2)  x² + 3x - 10
          -( x² - 2x)

            x                     
 x - 2)  x² + 3x - 10
          -x² + 2x
       ______________
                   5x  

Bring down -10

          x                         
 x - 2)  x² + 3x - 10
          -x² + 2x
       ______________
                   5x  - 10

The leading term of the new dividend (5x - 10) is 5x and the leading term of the divisor (x - 2) is x

5x ÷ x = 5

Write 5 as the second term of the quotient. Since 5 is positive, you can put a + sign between the first term and the second term.

        x  +  5               
 x - 2)  x² + 3x - 10
          -x² + 2x
       ______________
                   5x  - 10

Multiply the second term of the quotient by the divisor

5(x - 2) = 5x - 10

Subtract 5x - 10 from 5x - 10

          x  +  5                     
 x - 2)  x² + 3x - 10
          -x² + 2x
        ______________
                   5x  - 10
                  -5x + 10
              __________
                        0

(x² + 3x - 10) ÷ (x - 2) = x + 5

Another straightforward example showing how to do polynomial long division with remainders

Example #2

Divide x² - 5x + 1 by x + 3

Divide the leading term of x² - 5x + 1 by the leading term of x + 3

x² ÷ x = x

Write x as the first term of the quotient 

          x                         
 x + 3)  x² - 5x + 1

Multiply the first term of the quotient by the divisor

x(x + 3) = x² + 3x

Subtract x² + 3x from the dividend

          x                         
 x + 3)  x² - 5x + 1
          -( x² + 3x)

          x                         
 x + 3)  x² - 5x + 1
          -x² - 3x
       ______________
                -8x  

Bring down 1

          x                         
 x + 3)  x² - 5x + 1
          -x² - 3x
       ______________
                   -8x  + 1

Divide the leading term of -8x  + 1 by the leading term of x + 3

-8x ÷ x = -8

Write -8 as the second term of the quotient. Notice that putting a + sign between the first term and the second term does not change the problem.

            x + -8                     
 x + 3)  x² - 5x + 1
          -x² - 3x
       ______________
                   -8x  + 1

Multiply the second term of the quotient by the divisor

-8(x + 3) = -8x - 24

Subtract -8x - 24 from -8x + 1

          x  + -8                     
 x + 3)  x² - 5x + 1
         -x² - 3x
        ______________
                   -8x  + 1
                    8x + 24
              __________
                           25       

(x² - 5x + 1) ÷ (x + 3) = x + -8 with a remainder of 25

What can polynomial long division be used for?

You can use polynomial long division to find the factors of a polynomial and as a result solve the polynomial. 

If the remainder is zero, then the quotient and divisor are factors of the polynomial. Since example #1 gives a zero remainder, x - 2 and x + 5 are factors of the polynomial x² + 3x - 10.

x² + 3x - 10 = (x - 2)(x + 5)

Since example #2 does not give a zero remainder, x + 3 and x - 8 are not factors of the polynomial x² - 5x + 1.

The division algorithm for polynomials

Suppose f(x) and d(x) are polynomials, with d(x) not equal to zero and the degree of d(x) is smaller or equal to the degree of f(x). Then, there exist a unique polynomial r(x) and a unique polynomial q(x) such that

f(x) = d(x) × q(x) + r(x)

• f(x) = dividend
• d(x) = divisor
• q(x) = quotient
• r(x) = remainder

r(x) is either equal to zero or it is of degree less than the degree of d(x). If r(x) = 0, we say that d(x) divides evenly into f(x) and that d(x) and q(x) are factors of f(x).