A polynomial is an algebraic expression having multiple terms. For example, the expression x ^4 + x^3 -7 is a polynomial having 3 terms.  It is also known as trinomial because it has 3 terms. Similarly, x ^2 - 1 is a binomial expression because it has 2 terms. Binomials and trinomials are polynomial functions because they contain more than one term unlike monomials. A monomial is not a polynomial because it does not satisfy the basic definition of  a polynomial.

 

Adding and subtracting polynomials is easy because we just combine the like terms and apply the arithmetic operations of addition and subtraction. On the other hand, the division of polynomial can be tricky. We can divide two polynomials expressions either by using long division or by using synthetic division. In algebra, a long division method is used for dividing one polynomial by another. The polynomial which is a divisor in long division is either of the same degree or of the lower degree than the polynomial in the dividend.

Polynomial long division method can be done manually following step by step instructions and rules. The length of the division depends upon the degree of a polynomial.

The expression used to depict the outcome of the polynomial division is given below:

A = BQ + R

Here, A is the dividend, B is the divisor, Q and R are quotient and remainder respectively.

A polynomial yields a zero remainder if the divisor B is the factor of the dividend A. Before solving polynomial division problems, we need to arrange the polynomial function in the dividend in the descending order. In this article, we will discuss how to divide polynomials using the long division method through different examples. So, let us get started.

Example 1

Divide x ^5 - 2x ^4 - x^3 + 3x - 1 by x - 1.

Solution

In the above division problem, x ^5 - 2x ^4 - x^3 + 3x - 1 is the dividend and x - 1 is the divisor.  The highest degree of the polynomial is 5. We repeat the process in step 1 again and again until we reach the remainder.

Follow these step by step instructions to divide the polynomials and find the quotient.

Step 1

  • Write the polynomial in the form x ^5 - 2x ^4 - x^3 + 0x ^2 +3x - 1
  • We will start by dividing the first term of the dividend by the first term of the divisor. \frac {x ^5 } {x} = x ^4. Write this answer above at the quotient's place. It is the first term of the quotient.
  • Now, we need to multiply x ^4 with the divisor x - 1. x ^4 \cdot x - 1 = x ^5 - x^4.
  • Subtract it from the dividend x ^5 - 2x ^4 - x^3 + 0x ^2 + 3x -1 - x ^5 + x^ 4 = -x^4 - x^3 + 0x^2 + 3x - 1. This is the remaining part of the dividend.

Step 2

  • Divide the first term of the remaining dividend in the step 1 by the first term of the divisor. \frac {-x^4}{x} = -x^3. Write this result in the quotient's place. This is the second term of the quotient.
  • Now, multiplying -x ^3 by x - 1 will give us the following result.

-x ^3 \cdot x - 1 = -x ^4 + x ^3

  • Subtract it from the remaining dividend. -x^4 - x^3 + 0x^2 + 3x - 3 + x ^4 - x^3 = -2x ^3 +3x - 1. This is the remaining part of the dividend.

Step 3

  • Divide the leading term of the remaining part of the dividend from step 2 by the first term of the divisor. \frac{-2x ^3}{x} = -2x^2. Write it at the quotient's place. This is the third term of the quotient.
  • Multiply -2x ^2 by the divisor x - 1 to get -2x ^3 + 2x ^2.
  • Subtract it from the remaining dividend -2x ^3 +3x - 1 + 2x ^3 -2x ^2= -2x^2 + 3x - 1. This is the remaining part of the dividend.

Step 4

  • Divide the first term of the remaining dividend in step 3 by the first term of the divisor. \frac {-2x ^2 }{x} = -2x. Write it at the quotient's place. This is the fourth term of the quotient.
  • Multiply it with the divisor to obtain -2x ^2 + 2x.
  • Subtract it from the remaining dividend to get -2x^2 + 3x - 1 + 2x ^2 - 2x = x + 1. This is the remaining part of the dividend.

Step 5

  • Divide the first term of the divisor by the first term of the remaining dividend. \frac {x}{x} = 1. Write this result at the quotient's place. This is the fifth term of the quotient.
  • Multiply the divisor by 1 to get x + 1.
  • Subtract x + 1 from the remaining dividend x + 1 to get 0.
  • Hence, your division is complete.

Remember that when a polynomial is completely divisible by another polynomial, then it means that the divisor is the factor of the dividend. In the above example,  x - 1 is the factor of the polynomial x ^5 - 2x ^4 - x^3 + 3x - 1.

\begin{array}{ccccccccccc} & & &  x^4 & - x^3 & -2x^2 &-2x &+ 1&\\ \cline{3-10} \multicolumn{2}{r}{x-1 \surd} & x^5 &-2x^4 & -x^3&+0x ^2 &+3x &-1 && \\ & &x^5&-x^4& & & & \\ \cline{3-7} & & &-x^4&-x^3 &+0x ^2 &+3x &-1 & & \\ & & &-x^4&+x^3&& & & \\ \cline{4-8} & & & & -2x^3&+0x^2 &+3x&-1& & \\ & & & &-2x^3& +2x^2&& & \\ \cline{5-9} & & & & &-2x^2 &+3x& -1& \\ & & & & &-2x^2  &+2x& \\ \cline{6-10} & & & & & & x& -1 \\ & & & & & & x& -1 \\ \cline{6-10} & & & & & & &0&&\\ \end{array}

 

Superprof

Example 2

Divide x ^3 - 1 by x - 1

Solution

In the above example, x ^3 - 1  is the dividend and  x - 1 is the divisor. In other words, we can say that x ^3 - 1 is the numerator and x - 1 is the denominator. Follow these step by step instructions to obtain a quotient of the above example.

Step 1

  • It is better to write the dividend as x ^3 +0 x^2 + 0x - 1 for your ease of calculation.
  • Divide the first term of the dividend by the first term of the divisor. \frac {x ^3}{x} = x^2. This will be the first term of the quotient.
  • Multiply x ^2 with the divisor x - 1 to get x ^3 - x^2.
  • Subtract it from the dividend x ^3 +0 x^2 + 0x - 1 - x^3 + x^2 = x ^2 + 0x - 1. This is the remaining dividend.

Step 2

  • Divide the first term of the remaining dividend by the first term of the divisor. \frac {x ^ 2}{x} = x. This is the second term of the quotient.
  • Multiply x with x - 1 to get x ^2 - x.
  • Subtract it from the remaining part of the dividend to get x ^ 2 + 0x - 1 - x^2 + x= x - 1. We will cancel out x^2 and -x^2. x - 1 is the remaining part of the dividend in step 1.

Step 3

  • Divide the first term of the remaining dividend by the first term of the divisor. \frac{x - 1}{x} = +1. This is the third term of the quotient.
  • Multiply it with the divisor to get x - 1.
  • Subtract it from the remaining part of the dividend in step 2 to get 0.

Hence, our division is complete. Our quotient is x ^2 + x + 1 and our remainder is equal to 0 which means that the divisor in this example is the factor of the dividend.

\begin{array}{ccccccccccc} & & & x^2 & +x& +1&\\ \cline{3-7} \multicolumn{2}{r}{x-1 \surd} & x^3 &+0x^2 &+0x & -1& \\ & &x^3&-x ^2 & & & \\ \cline{3-7} & & &x^2&+0x &+ 1 & \\ & & &x^2&- x&& & & \\ \cline{4-8} & & & & x&-1 & & \\ & & & &x& -1& & \\ \cline{5-9} & & & & &0& \\ \end{array}

 

Example 3

Divide x ^2 - 3x + 1 by x + 1

Solution

In the above example, x ^2 - 3x + 1 is the dividend and  x + 1 is the divisor. Follow these step by step instructions to solve the above example by polynomial long division method.

Step 1

  • Divide the first term of the dividend by the first term of the divisor. \frac { x ^2 }{x} = x. This is the first term of the quotient.
  • Multiply the divisor with this first term of the quotient to get x + 1 \cdot x = x ^2 + x.
  • Subtract it from the dividend to get x ^2 - 3x + 1 - x^2 - x = -4x + 1. This is the remaining part of the dividend.

Step 2

  • Divide the first term of the remaining dividend by the first term of the divisor. \frac{-4x}{x} = -4. This is the second term of the quotient.
  • Multiply it by the divisor to get x + 1 \cdot -4 = -4x - 4.
  • Subtract it from the remaining part of the dividend from the step 1. -4x + 1 +4x +4 = 5.

Hence, the division is complete because 5 cannot be further divided by the dividend. Hence, 5 is the remainder of this division. Since we did not get 0 as the remainder of this example, so it means that the divisor is not the factor of the dividend.

 

\begin{array}{ccccccccccc} & & & x & -4&\\ \cline{3-7} \multicolumn{2}{r}{x+1 \surd} & x^2 &-3x &+1& \\ & &x^2&+x & & & \\ \cline{3-7} & & &-4x&+1 & \\ & & &-4x&-4& & & \\ \cline{3-7} & & & &5 & & \\ \end{array}

Example 4

Divide x ^4 + x + 1 by x + 1.

Solution

In the above example, x ^4 + x + 1 is the dividend and x + 1 is the divisor. Follow these step by step instructions to solve the above example through polynomial long division method.

Step 1

  • Write the dividend as x ^4 + 0x ^3 + 0x^2+ x + 1 for the ease of your calculation.
  • Divide the first term of the dividend by the first term of the divisor. \frac{x ^4}{x} = x^3. This is the first term of the quotient.
  • Multiply it by the divisor. x ^3 \cdot x + 1 = x^4 + x^3.
  • Subtract it from the dividend. x ^4 + 0x^3 + 0x ^2 + x + 1 - x ^4 - x^3 = - x^3 + 0x ^2 + x + 1. This is the remaining part of the dividend.

Step 2

  • Divide the first term of the remaining dividend in step 1 by the first term of the divisor. \frac {-x ^3}{x} =-x ^2. This is the second term of the quotient.
  • Multiply it by the divisor. -x ^2 \cdot x + 1 = -x ^3 - x^2.
  • Subtract it from the remaining part of the dividend in the step 1. -x ^3 + 0x ^2 + x + 1 + x ^3 + x^2= x^2 + x + 1. This is the remaining part of the dividend.

Step 3

  • Divide the first term of the remaining dividend in step 2 by the first term of the divisor. \frac{x ^2}{x} = x. This is the third term of the quotient.
  • Multiply it by the divisor. x \cdot (x + 1) = x ^2 + x.
  • Subtract it from the remaining part of the dividend in the step 2. x^2 + x + 1 - x^2 - x = 1.

The division will end here because 1 cannot be divided further by the divisor. Hence, 1 is the remainder of this example.

\begin{array}{ccccccccccc} & & & x ^3 & -x ^2&+x&\\ \cline{3-7} \multicolumn{2}{r}{x+1 \surd} & x^4 &+0x ^3 & +0x^2&+0x&+1& \\ & &x^4&+x^3& & & \\ \cline{3-7} & & &-x^3&+0x^2&+x&+1 & \\ & & &-x^3&-x^2& & & \\ \cline{3-7} & & &&x^2&+x&+1 & \\ & & &&x^2&+x& & & \\ \cline{3-7} & & & &&&1 & & \\ \end{array}

 

Example 5

Divide 2x^2 + x - 1 by 2x - 1.

Solution

In the above example, 2x^2 + x - 1 is the dividend and 2x - 1 is the divisor. Follow these step by step instructions to solve this example by employing polynomial long division method.

Solution

Step 1

  • Divide the first term of the dividend by the first term of the divisor. \frac {2x ^2}{2x} = x. This is the first term of the quotient.
  • Multiply it by the divisor. (2x - 1) \cdot x = 2x^2 - x.
  • Subtract it from the dividend. 2x^2 + x - 1 - 2x ^2 + x = 2x - 1. This is the remaining part of the dividend.

Step 2

  • Divide the first part of the remaining dividend obtained in step 1 by the first term of the divisor. \frac{2x}{2x} = 1. This is the second term of the quotient.
  • Multiply it by the divisor. (2x - 1) \cdot 1 = 2x - 1.
  • Subtract it from the remaining part of the dividend from step 1 to get 0.

Hence, the division is complete. Remember that when a polynomial is completely divisible by another polynomial, then it means that the divisor is the factor of the dividend. In the above example,  2x - 1 is the factor of the polynomial 2x^2 + x - 1.

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.

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