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# Dividing Polynomials – Explanation & Examples

**Division of polynomials** might seem like the most challenging and intimidating of the operations to master, but so long as you can recall the basic rules about the long division of integers, it’s a surprisingly easy process.

In this article, we are going to learn **how to carry out the division between two monomials, a monomial and polynomial and lastly between two polynomials.**

Before getting into this topic of dividing polynomials, let’s briefly discuss a few important terms here.

**Polynomial**

A **polynomial is an algebraic expression made up of two or more terms which are subtracted, added or multiplied**. A polynomial can contain coefficients, variables, exponents, constants and operators such addition and subtraction.

It is also important to note that, a polynomial can’t have fractional or negative exponents. Examples of polynomials are; 3y^{2} + 2x + 5, x^{3} + 2 x ^{2} − 9 x – 4, 10 x ^{3} + 5 x + y, 4x^{2} – 5x + 7) etc.

*There are three types of polynomials, namely monomial, binomial and trinomial.*

**Monomial**

A monomial is an algebraic expression with only one term. Examples of monomials are; 5, 2x, 3a^{2}, 4xy etc.

**Binomial**

A binomial is an expression containing two terms which are separated by either addition sign (+) or subtraction sign (-). Examples of binomial expressions are 2*x* + 3, 3*x* – 1, 2x+5y, 6x−3y etc.

**Trinomial**

A trinomial is an expression that contains exactly three terms. Examples of trinomials are:

4x^{2} + 9x + 7, 12pq + 4x^{2} – 10, 3x + 5x^{2} – 6x^{3 }etc.

## How to Divide Polynomials?

The division is an arithmetic operation of splitting a quantity into equal amounts. The division process is sometimes referred to as repeated subtraction or the reverse multiplication.

*There are two methods in mathematics for dividing polynomials. *

These are the long division and the synthetic method. As the name suggests, the long division method is most cumbersome and intimidating process to master. On the other hand, the **synthetic method** is a “**fun**” way of dividing polynomials.

### How to divide a monomial by another monomial?

When dividing a monomial by another monomial, we divide the coefficients and apply the quotient law x ^{m} ÷ x ^{n} = x ^{m – n} to the variables.

* NOTE:* Any number or variable raised to the power of zero is 1. For example, x

^{0 }= 1.

Let’s try a few examples here.

*Example 1*

Divide 40x^{2} by 10x

__Solution__

Divide the coefficients first

40/10 = 4

Now divide the variables using the quotient rule

x^{2} /x = x^{2 -1}

= x

Multiply the quotient of the coefficients by the quotients of the variables;

⟹ 4* x = 4x

Alternatively;

40x^{2}/10x = (2 * 2 * 5 * 2* x * x)/ (2 * 5 * x)

Since x, 2 and 5 are common factors of both the denominator and numerator, we cancel them out to get;

⟹ 40x^{2}/10x = 4x

*Example 2*

Divide -15x^{3}yz^{3} by -5xyz^{2}

__Solution__

Divide the coefficients normally and use the quotient law x ^{m} ÷ x ^{n} = x ^{m – n} to divide the variables.

-15x^{3}yz^{3} / -5xyz^{2} ⟹ (^{-15}/_{-5}) x^{3 – 1}y^{1 – 1}z^{3 – 2}

= 3 x^{2}y^{0}z^{1}

= 3x^{2}z.

*Example 3*

Divide 35x^{3}yz^{2} by -7xyz

__Solution__

Using the Quotient law

35x^{3}yz^{2} / -7xyz ⟹ (^{35}/_{-7}) x^{3 – 1}y^{1 – 1}z^{2 – 1}

= -5 x^{2}y^{0}z^{1}

= -5x^{2}z.

*Example 4*

Divide 8x^{2}y^{3} by -2xy

__Solution__

8x^{2}y^{3}/-2xy ⟹ (^{8}/_{-2}) x^{2 – 1}y^{3 – 1}

= -4xy^{2}.

### How to divide polynomials by monomials?

To divide a polynomial by a monomial, separately divide each term of the polynomial by the monomial and add the quotient of each operation to get the answer.

Let’s try a few examples here.

*Example 5*

Divide 24x^{3} – 12xy + 9x by 3x.

__Solution__

(24x^{3}–12xy + 9x)/3x ⟹ (24x^{3}/3x) – (12xy/3x) + (9x/3x)

= 8x^{2 }– 4y + 3

*Example 6*

Divide 20x^{3}y + 12x^{2}y^{2} – 10xy by 2xy

__Solution__

(20x^{3}y + 12x^{2}y^{2} – 10xy) /(2xy) ⟹ 20x^{3}y /2xy + 12x^{2}y^{2}/2xy – 10xy/2xy

= 10x^{2} + 6xy – 5.

*Example 7*

Divide x^{6} + 7x^{5} – 5x^{4} by x^{2}

__Solution__

= (x^{6} + 7x^{5} – 5x^{4})/ (x^{2}) ⟹ x^{6} /x^{2 +} 7x^{5}/x^{2} – 5x^{4}/x^{2}

Use the Quotient law to divide the variables

= x^{4 }+ 7x^{3 }− 5x^{2}

*Example 8*

Divide 6x^{5} + 18x^{4} – 3x^{2} by 3x^{2}

__Solution__

= (6x^{5} + 18x^{4} – 3x^{2})/3x^{2} ⟹ 6x^{5}/3x^{2} + 18x^{4}/3x^{2} – 3x^{2}/3x^{2}

=2x^{3} + 6x^{2} – 1.

*Example 9*

Divide 4m^{4}n^{4} – 8m^{3}n^{4} + 6mn^{3} by -2mn

__Solution__

= (4m^{4}n^{4} – 8m^{3}n^{4} + 6mn^{3})/(-2mn) ⟹ 4m^{4}n^{4}/- 2mn – 8m^{3}n^{4}/-2mn + 6mn^{3}/-2mn

= 2m^{3}n^{3} + 4m^{2}n^{3} – 3n^{2}

*Example 9*

Solve (a^{3} – a^{2}b – a^{2}b^{2}) ÷ a^{2}

__Solution__

= (a^{3} – a^{2}b – a^{2}b^{2}) ÷ a^{2 }⟹ a^{3}/ a^{2}– a^{2}b/ a^{2} – a^{2}b^{2}/ a^{2}

= a – b – b^{2}

### How to do polynomial long division?

The long division is the most suitable and reliable method of dividing polynomials, even though the procedure is a bit tiresome, the technique is practical for all problems.

The process of dividing polynomials is just similar to dividing integers or numbers using the long division method.

*To divide two polynomials, here are the procedures:*

- Arrange the both the divisor and dividend in descending order of their degrees.
- Divide the 1
^{st}term of the dividend by the 1^{st}term of divisor to obtain the 1^{st}term of the quotient. - Find the product of all the terms of the divisor and by the 1
^{st}term quotient and subtract the answer from the dividend. - If there is a remainder in the above, proceed as repeat procedure 3 until till you get zero as the remainder or you obtain an expression having a smaller degree than that of the divisor.

*Example 10*

Divide the following polynomials using long division method:

3x^{3} – 8x + 5 by x – 1

__Solution__

*Example 11*

Divide 12 – 14a² – 13a by 3 + 2a.

__Solution__

*Example 12*

Divide the polynomials below:

10x⁴ + 17x³ – 62x² + 30x – 3 by (2x² + 7x – 1).

__Solution__

*Practice Questions*

Divide the following polynomials:

- 20x by 5x
- 50x
^{5}y^{2}by10x^{4}y^{2} - 4x
^{3}– 6x^{2}+ 3x – 9 by 6x. - 6x
^{4}– 8x^{3}+ 12x – 4 by 2x^{2}. - 18xy + 22x
^{3}y -15xy^{2}by 3xy^{2} - 24x
^{2}y^{2 }-16x^{2}y -12xy^{3}by – 6x^{2}y^{2} - 4a
^{3}– 10a^{2}+ 5a by 2a - a
^{2}+ ab – ac by –a - 2x² + 3x + 1 by x + 1
- x² + 6x + 8 by x + 4
- 29x – 6x² – 28 by 3x -4).
- (
*x*^{3}+ 5*x*^{2}– 3*x*+ 4) by (*x*^{2}+ 1). - 5x
^{3 }– x^{2 }+6 by x – 4 - 4x
^{4 }−10x^{2 }+ 1 by x – 6 - 2x
^{3 }−3x − 5 by x + 2 - 9x
^{2}y + 12x^{3}y^{2}– 15xy^{3}by 6xy

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