Divide polynomials by monomials are arithmetic and mathematical operation that involves dividing a polynomial by a monomial, also known as a one-term polynomial. To divide polynomials by a monomial, multiply each term by the monomial. Determine the quotient: Divide each numerator term by the denominator. Each fraction should be simplified. In this article, we’ll look at various ways of splitting polynomials by monomials, definitions, and solved examples. For more such 7th Grade Math Concepts stay connected to us and resolve all your doubts.

## Dividing Polynomials by Monomials Rules

The principles for dividing polynomials by monomials are as follows:

- Divide the polynomial terms individually by the monomial.
- To get the cheapest form of each word, divide the coefficients and variables with powers individually.

Combine the results of the previous step that use the operator (+) or (-) depending on the sign of the words.

## How do you Divide Polynomial by a Monomial?

The most popular approach for dividing a polynomial by monomials is to split the polynomial terms split with (+) / (-) and solve each term independently. The combined result of all individual outcomes will be the final result. The factorization method is yet another way to split a polynomial by monomials. Let’s take a quick look at both ways, as mentioned below.

### Division of Polynomials by Monomials using Divide the Terms Method

In this procedure, we will divide the polynomial terms divided by the operators (+) / (-) and simplify each term. Let’s look at an example to get an in-depth clarification. As an example, (6y^{2} + 3y) / (3y) may well be done by following the steps outlined below for division polynomial by a monomial:

- Divide the polynomials 6y
^{2}+ 3y into its terms. 6y^{2}and 3y are the words. - Now divide each word by the monomial 3y, i.e., (6y
^{2}/3y) + (3y/3y). - Each term will be reduced to its simplest form by removing the common variables,

i.e., 6y^{2}/3y = 2y and 3y/3y = 1. - The separate results are now merged using the (+) operator. As a consequence, the answer is 2y + 1.
- As a result, the result of (6y
^{2}+ 3y) / (3y) Equals 2y + 1.

**Example:**

Using the dividing the term approach, divide the polynomials 16z^{3} – 20z^{2} by the monomials 4z^{2}?

**Solution:**

We will utilize the splitting the word approach to divide polynomials by monomials.

(16z^{3} – 20z^{2}) / 4z^{2}

= (16z^{3}/4z^{2}) – (20z^{2}/4z^{2})

= 4z – 5

As a result, the result of (16z^{3} – 20z^{2}) / 4z^{2} is 4z – 5.

**Points to Remember**

To reduce the divisions of algebraic expressions, we divide the coefficients and variables with powers. To simplify 6y^{2}/3y, first, examine the coefficient values that really are 6/3 = 2, and then consider the components with exponents as y^{2}/y = y. The combined result is now 2y. As a result, 6y^{2}/3y = 2y.

### Dividing Polynomials by Monomials using Factorization Method

When using the factorization technique to divide polynomials by monomials, we first discover the common element between both the numerator and denominator. Let’s look at an example to get a clear idea. For example, to factorize (8x^{2} + 4x) 4x, we can use the following procedures for division polynomial by a monomial:

- We’ll focus on common variables between both the numerator and the denominator. We can see that the common factor of (8x
^{2}+ 4x) and 4x is 4x. - Thus, the statement may be expressed as 4x(2x + 1) / 4x by removing the common component 4x from the parenthesis and keeping the denominator 4x.
- When we remove the common phrase 4x, we obtain the solution as 2x + 1.
- As a consequence, the outcome of (8x
^{2}+ 4x) 4x is 2x + 1.

### Dividing Polynomial by Monomial Examples

**1. Divide x ^{6} + 7x^{5} â€“ 5×4 by x^{2}?**

= x^{6} + 7x^{5} â€“ 5×4 Ã· x^{2}

Now we must divide each polynomial term even by monomial and simplify.

= x^{6} /x^{2}+ 7x^{5} /x^{2}â€“ 5×4/ x^{2}

Each word will now be simplified by removing the common element.

= x^{4}+7x^{3}âˆ’5x^{2}

**2. Divide a ^{2} + ab â€“ ac by â€“a?**

= a^{2} + ab â€“ ac Ã· -a

= a^{2} + ab â€“ ac /-a

Now we must divide each polynomial term even by monomial and simplify.

= a^{2}/-a+ ab/-a â€“ ac /-a

= -a^{2}/a-ab/a + ac /a

Each word will now be simplified by removing the common element.

= -a – b + c

**3. Find the quotient 4m ^{4}n^{4} â€“ 8m^{3}n^{4} + 6mn^{3} by -2mn?**

= 4m^{4}n^{4} â€“ 8m^{3}n^{4} + 6mn^{3} Ã· -2mn

Now we must divide each polynomial term even by monomial and simplify.

= 4m^{4}n^{4} /-2mn â€“ 8m^{3}n4 /-2mn + 6mn^{3} /-2mn

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

Each word will now be simplified by removing the common element.

= 2m^{3}n^{3} + 4m^{2}n^{3} – 3n^{2}

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### FAQs on Dividing Polynomials by Monomials

**1. How do you Divide Polynomials by Monomials?**

Polynomial can be divided by monomial by dividing the polynomial’s terms and dividing each one by the monomial separately before linking them using the operator (+) / (-) depending on the value of the terms. Division (5x^{2} – 15x) / 5x for example. = (5x^{2}/5x) – (15x/5x) = x â€“ 3

**2. What is the first rule in Dividing Polynomials by Monomials?**

The first principle for dividing polynomials by monomials is to Divide the polynomial terms individually by the monomial.

**3. What is the best method for splitting polynomials by monomials for you, and why?**

Long division is the preferred approach for dividing polynomials by monomials since it includes divisions that leave remainders. As a result, it aids us in understanding the quotient and remainder.