Dividing Polynomials | Long Division | Synthetic Division | Factorization Methods

Dividing Polynomials in maths is an arithmetic operation in which one polynomial is divided by another polynomial, where the divisor polynomial must have a degree less than or equal to the Dividend Polynomial otherwise division of polynomial can't take place. The most general form of a polynomial is given as:

a_nxⁿ + a_n−1xⁿ⁻¹ + ... + a₂x² + a₁x + a₀

Where a₀, a₁, a₂, . . ., a_n are the real coefficients. In Dividing Polynomial we divide the polynomial with a higher degree by a polynomial (that can be a monomial, binomial, trinomial, or any other higher degree polynomial) with less degree.

There are various methods of Dividing Polynomials, some of those methods are:

• Long Division
• Synthetic Division
• Polynomial Division Using Factors

Long Division of Polynomials

The long division method is the most frequent and general method for dividing polynomials by binomials or any other form of polynomials. In case the given numerator and denominator do not have any common factors, you can simplify the expression by using the long division method.

Steps to Divide Polynomials using Long Division Method

To divide the polynomial using Long Division, we can use the following steps:

Step 1: Arrenge both Divisors and Dividends in the decreasing order of degree of each of term i.e., a_nxⁿ + a_n−1xⁿ⁻¹ + . . . + a₁x + a_0.

Step 2: Arrenge the Divisor and Dividend Long Division Form.

Step 3: Divide the the dividend's first term(xⁿ) by the divisor's first term, and use it as the quotient's first term.

Step 4: Multiply the divisor by the result of step 2 and arrenge them under the divident such that like terms aligned with each other.

Step 5: Subtract the result of step 3 from the divisor to create new polynomial.

Step 6: Repeat the steps 3, 4, 5 untill the resulting polynomial has a degree less then divisor.

Examples of Dividing Polynomials Using Long Division: Divide x² + 2x + 3 by x - 2.

Solution:

Step 1: Arrange both Divisors and Dividends in the decreasing order of degree

Dividend = x² + 2x + 3
Divisor = x - 2

Step 2: We will write the dividend and divisor in the long division form, like this:

Step 3: Divide the first term of the dividend by the first term of the divisor, to get the first term of divisor i.e.,

x²/x = x

This means that the first term of the quotient is x.

Step 4: Multiply the divisor (x - 2) by the first term of the quotient (x), and arrenge them under the divident with like terms aligned.

Step 5: Subtracted the result of step 3 from the dividend to get a new polynomial,

Step 6: We will repeat steps 3 and 4 using the new polynomial 4x + 3:

4x/x = 4

Multiplying divisor (x - 2) by the 4,

Step 7: Since the degree of the new polynomial 11, has a degree 0 which is less than the degree of the divisor (x - 2) i.e., 1, we can stop here.

So, the result of dividing the polynomial x² + 2x + 3 by x - 2 is:

Quotient: x + 4 and Remainder: 11.

Example: Divide x³ + 2x² - 5x + 1 by x - 2.

Solution:

Synthetic Division of Polynomials

It is a technique for dividing a polynomial by a linear binomial using only the coefficient values. We write the polynomials in standard form from the greatest degree term to the lowest degree term in this manner. Use zero as the coefficients of the missing terms when writing in descending powers.

Steps to Divide Polynomials Using Synthetic Division

To divide the polynomials using synthetic division, we can use the following steps:

Step 1: Write the polynomial in standard form, with the terms arranged in descending order of degree.

Let's Suppose a polynomial as a₃x³ + a₂x² + a₁x + a_0.

Step 2: Identify the divisor, and ensure that the divisor is in the form of x − k

Step 3: Write the value k to the left of the division symbol and place a vertical line to the right of it, and write the coefficients of the polynomial to be divided in the spaces to the right of the vertical line. Where any missing coefficients should be represented by zeros. 

Step 4: Bring down the first coefficient and write it below the line, and multiple it with k and write the result in the next space to the right.

Step 5: Add both values in the second space and wrote it in space next to the leading coefficient.

Step 6: Multiply the result of Step 5 with k and write it in the third space, and add both column values to get the further result.

Step 7: Repeat this till reach the real constant part.

If a₀ + k(a₁ + k(a₂ + ka₃)) = 0, then x − k is a factor of f(x). Otherwise, the remainder of the division is a₀ + k(a₁ + k(a₂ + ka₃)).

Step 8: The resulting values represent the coefficient of the remainder (last value) and  qotient (other values).

Example: Divide 3x⁴ + 5x + 9 by x – 1 using Synthetic Division

Write the polynomial in standard form, with the terms arranged in descending order of degree: 3x⁴ + 0x³ + 0x² + 5x + 9,
Identify the divisor, and ensure that the divisor is in the form of x − k: Divisor = x – 1,

Then follow these steps:

Step 1:	Step 2:
Step 3:	Step 4:
Step 5:	Step 6:

The resulting values represent the coefficient of the remainder (last value) and quotient (other values).

Therefore, the quotient is 3x³ + 3x²+ 3x + 8 and the remainder is 17.

Example: Divide the polynomial x² + x - 2 by x - 1 by synthetic division.

Solution:

\begin{array}{c|rrr}&1&1&-2\\1&&1&2\\\hline\\&1&2&0\\\end{array}

The first two numbers of the last row represent the coefficients of the quotient and the third value is the remainder.

Thus, the quotient is x + 2 and remainder is 0.

Dividing Polynomial by Monomial

When a polynomial is divided by a monomial i.e., a polynomial with only one term in it, then the resulting polynomial can only be found if the degree of the divisor is less than or equal to the degree of the polynomial under consideration. We can divide a polynomial by a monomial using the following methods:

• Splitting the Terms Method
• Factorization Method

Let's understand these methods in detail as follows:

Dividing Polynomial Using Splitting the Term Method

This method involves splitting each term of the polynomial into separate terms and then simplifying them by dividing each term with the monomial. Let's consider an example to understand this method better.

Example: Divide the polynomial 6x³ + 12x² + 9x by the monomial 3x.

Solution:

Step 1: Split the polynomial into separate terms.

(6x³ + 12x² + 9x) ÷ 3x = (6x³)/(3x) + (12x²)/(3x) + (9x)/(3x)

Step 2: Simplify each term by dividing with the monomial 3x:

(6x³ + 12x² + 9x) ÷ 3x = 2x² + 4x + 3

Therefore, 6x³ + 12x² + 9x divided by 3x is equal to 2x² + 4x + 3.

Dividing Polynomial Using Factorization Method

This method involves factoring out the monomial from each term of the polynomial and then simplifying the expression. Let's consider an example to understand this method better.

Example: Divide the polynomial 15x³ - 25x² + 10x by the monomial 5x.

Solution:

Step 1: Factor out the monomial 5x from each term of the polynomial.

15x³ - 25x² + 10x = 5x(3x² - 5x + 2)

Step 2: The other factor than 5x is the required answer to the division.

5x(3x² - 5x + 2)/5x = 3x² - 5x + 2

Therefore, 15x³ - 25x² + 10x divided by 5x is equal to 3x² - x + 2.

Examples of Dividing Polynomial by Monomial

Let's consider some more examples to further understand the concept of dividing a polynomial by a monomial.

Example 1: Divide the polynomial 2x² + 6x + 4 by the monomial 2x.

Solution:

(2x² + 6x + 4) ÷ 2x = (2x²)/(2x) + (6x)/(2x) + (4)/(2x)

(2x² + 6x + 4) ÷ 2x = x + 3 + 2/x

Therefore, 2x² + 6x + 4 divided by 2x is equal to x + 3 + 2/x.

Example 2: Divide the polynomial 9x³ - 15x² + 6x by the monomial 3x.

Solution:

9x³ - 15x² + 6x = 3x(3x² - 5x + 2)

Therefore, 9x³ - 15x² + 6x divided by 3x is equal to 3x² - 5x + 2.

Dividing Polynomial by Binomial using Factorization

Factorization is the method of writing the given polynomial into product of its factors. If the binomial by which we are dividing a given polynomial is the factor of the given polynomial then we can eliminate the binomial and we are left with the other factor as the quotient. This can be better understood by the example given below:

Example: Divide x² + 4x + 3 by x + 1

Solution:

We have the Polynomial x² + 4x + 3 as dividend

The binomial x + 1 is our divisior

We will factorize the polynomial x2 + 4x + 3 using Middle Term Splitting method

⇒ x² + 4x + 3 = x² + 3x + x + 3 = x² + 3x + x + 3 = x(x + 3) + 1(x + 3) = (x + 1)(x + 3)

Now we can do the division as follows

(x² + 4x + 3)/(x + 1) = (x + 1)(x + 3)/(x + 1)

Now we will eliminate the common part (x + 1).

Hence we are left with (x + 3) as the quotient.

Common Mistakes to Avoid When Dividing Polynomials

There are some common mistakes done by students when dividing one polynomial by another. Some of these common mistakes are as follows:

• Making an error when performing the division: Double-check your arithmetic when performing the division. It is easy to make a mistake when working with long polynomials, so take your time and be thorough.
• Misaligning the terms when setting up the division: Make sure to align the terms correctly when setting up the division. Each term should be in the correct position relative to the other terms in the dividend and divisor.
• Forgetting to include the remainder: When dividing polynomials, there may be a remainder. Make sure to include the remainder in your answer, if there is one, and if the remainder is 0 then write 0 in the place of the remainder.
• Not simplifying the answer: Finally, simplify the answer as much as possible. This means combining like terms and putting the polynomial in standard form.

People Also Read:

• Degree of Polynomial
• Zeros of Polynomial
• Algebraic Identities of Polynomials

Sample Problems on Dividing Polynomials

Problem 1. Using synthetic division, find the quotient and remainder of \bold{\frac{x^2 + 3}{x - 4}} 

Solution:

Dividend = x² + 3 or, x² + 0x + 3

Divisor = x - 4

Applying synthetic division, we have:

\begin{array}{c|rrr}&1&0&3\\4&&4&16\\\hline\\&1&4&19\\\end{array}

The first two numbers of the last row represent the coefficients of the quotient and the third value is the remainder.

Thus, the quotient is x + 4 and the remainder is 19.

Problem 2. Solve \bold{\frac{4x^3+5x^2+5x+8}{4x+1}}using long division.

Solution:

Dividend = 4x³ + 5x² + 5x + 8

Divisor = 4x + 1

Using long division method, we have:

\begin{array}{r} x^2+x+1\phantom{)}   \\ 4x+1{\overline{\smash{\big)}\,4x^3+5x^2+5x+8\phantom{)}}}\\ \underline{4x^3~\phantom{}+x^2~~~~~~~~~~~\phantom{-b)}}\\ 4x^2+5x~~~~~~~\phantom{)}\\ \underline{~\phantom{()}4x^2+1x~~~~~~~~~}\\ 4x+8\phantom{)}\\ \underline{-~\phantom{()}(4x+1)}\\ 7\phantom{)}\\ \end{array}

Thus, the quotient and remainder are x² + x + 1 and 7 respectively.

Problem 3. Solve \bold{\frac{4x^3-3x^2+3x-1}{x-1}} using synthetic division.

Solution:

Dividend = 4x³ - 3x² + 3x - 1

Divisor = x - 1

Applying synthetic division, we have:

\begin{array}{c|rrr}&4&-3&3&-1\\1&&4&1&4\\\hline\\&4&1&4&3\\\end{array}

The first three numbers of the last row represent the coefficients of the quotient and the fourth value is the remainder.

The quotient is 4x² + x + 4 and the remainder is 3.

Problem 4. Solve \bold{\left( 5{{x}^{3}}-6{{x}^{2}}+3x+11 \right)\div \left( x-2 \right)} using synthetic division.

Solution:

Dividend = 5x³ - 6x² + 3x + 11

Divisor = x - 2

Applying synthetic division, we have:

\begin{array}{c|rrr}&5&-6&3&11\\2&&10&8&22\\\hline\\&5&4&11&33\\\end{array}

The first three numbers of the last row represent the coefficients of the quotient and the fourth value is the remainder.

The quotient is 5x² + 4x + 11 and the remainder is 33.

Problem 5. Solve \bold{\left( 18{{x}^{4}}+9{{x}^{3}}+3{{x}^{2}} \right)\div \left( 3{{x}^{2}}+1 \right) } using long division.

Solution:

Dividend = 18x⁴ + 9x³ + 3x² + 0x + 0

Divisor = 3x² + 1

Using long division method, we have:

\begin{array}{r} 6x^2+3x-1\phantom{)}   \\ 3x^2+1{\overline{\smash{\big)}\,18x^4+9x^3+3x^2+0x+0\phantom{)}}}\\ \underline{18x^4~\phantom{}+0x^3+6x^2~~~~~~~~~\phantom{-b)}}\\ 9x^3-3x^2+0x+0\phantom{)}\\ \underline{~\phantom{()}9x^3+0x^2+3x~~~~~~~~~}\\ -3x^2-3x+0\phantom{)}\\ \underline{~\phantom{()}-3x^2+0x-1}\\ -3x+1\phantom{)}\\ \end{array}

Thus, the quotient and remainder are 6x² + 3x - 1 and -3x + 1 respectively.

Unsolved Practice Questions on Dividing Polynomials

Question 1: Divide the polynomial x² + 5x + 6 by x + 3.

Question 2: Use long division to divide 3x³ - 2x² + 4x - 1 by x - 1.

Question 3: Divide 4x⁴ - 9x³ + 5x² + x - 2 by 2x - 3.

Question 4: Use long division to divide x⁴ + 2x³ - x + 4 by x² - x + 1.

Question 5: Use synthetic division to divide 5x³ - 3x² + 2x + 7 by x - 2.

Question 6: Divide 5x3−4x2+3x−2 by x−2.

Question 7: Divide 2x3+3x2−x−1 by x+1.

Question 8: Divide 7x4−6x3+4x−5 by x2+1.

Question 9: Divide x3+3x2−4x−12 by x+3.

Question 10: Divide 8x3−4x2+2x−1 by 2x−1.