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. 

Polynomial long division

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

Step 1

First, make sure the divisor and dividend are in standard form. For example, if the dividend is 3x + x2 - 10, write it in standard form as x2 + 3x - 10.

Step 2

When you start, always divide the leading term of the dividend by the leading term of the divisor. This will give you the first term of the quotient.

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

Step 3

Multiply the first term of the quotient (x) by the divisor (x - 2) to get the product. Subtract the product from the dividend to get the difference. Then, bring down the next term, if any. The difference along with the term you brought down is your new dividend.

Notice that the new dividend is 5x - 10.

5x is the difference between x2 + 3x - 10 and x2 - 2x. 

-10 is the term you brought down

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

Step 4

At this point, the process will repeat itself.

Do step 2 with the new dividend (5x - 10). Divide the leading term of the new dividend by the leading term of the divisor. This will give you the second term of the quotient.

Do step 3. Multiply the second term of the quotient (5) by the divisor (x - 2) to get the product. Subtract the product from the new dividend to get the difference. Then, bring down the next term, if any. The difference along with the term you brought down is your "newest dividend."

Now, when exactly do you stop the process? Continue the process until the newest dividend or remainder is zero or when the degree of the newest dividend or remainder is smaller than the degree of the divisor. In many cases, you can achieve this when you end up with a remainder that has no variable.

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

In many cases, when the dividend is in standard, there can be some missing terms. What can you do then? Just add a term of that power with 0 as its coefficient.

For example, rewrite the third-degree polynomial  5x3 + 4x - 6 as 5x3 + 0x2 + 4x - 6 before doing long division.

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).

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