In Multiplication of Polynomial by Monomial, we will multiply each term of the polynomial by a monomial. We use distributive property and exponent rules while multiplying polynomials by monomials. We use some rules and steps to Multiply monomials by polynomials. To check all the guidelines, tricks, and processes, check out the 6th Grade Math Multiplying a Polynomial by a Monomial concept.

Also, check

- Multiplication of Two Monomials
- Worksheet on Multiplying Monomials
- Worksheet on Multiplying Monomial and Binomial

## Multiplying a Polynomial by Monomial – Definition

Multiplying a Polynomial with a Monomial will give you the product as a polynomial. Take the monomial and multiply every term of a polynomial with the monomial to find the product by Multiplying a Polynomial by a Monomial.

### How to Multiply Monomials and Polynomials?

Below is the process of multiplying a polynomial by a monomial.

(i) Firstly, identify the monomial and polynomial from the given expressions.

(ii) Write the monomial and the polynomial in the same row and then separate it by using the multiplication sign.

(iii) Apply the distributive law and multiply each term of the polynomial by a monomial.

(iv) Using exponent law, add the powers of the same variables.

(v) Then, simplify the resulting polynomial by adding or subtracting the like terms.

### Multiplying a Polynomial by a Monomial Examples

Various problems are solved in a simple manner included with tricks and explanations.

**Question 1.**

Multiply 6m^{2}n – 10mn^{2} + 8mn and 4mn

**Solution:**

From the given expressions, 4mn is the monomial, and 6m^{2}n – 10mn^{2} + 8mn is the polynomial.

Write the monomial and the polynomial in the same row and then separate them by using the multiplication sign.

4mn × (6m^{2}n – 10mn^{2} + 8mn)

Apply the distributive law and multiply each term of the polynomial by a monomial.

According to the distributive law, a × (b + c) = (a × b) + (b × c)

4mn × (6m^{2}n – 10mn^{2} + 8mn) = (4mn × 6m^{2}n) – (4mn × 10mn^{2}) + (4mn × 8mn)

Multiply the coefficients and variabes separately.

(4mn × 6m^{2}n) – (4mn × 10mn^{2}) + (4mn × 8mn) = (24m^{1 + 2}n^{1 + 1}) – (40m^{1 + 1}n^{1 + 2}) + (32m^{1 + 1}n^{1 + 1}) = 24m^{3}n^{2} – 40m^{2}n^{3} + 32m^{2}n^{2}

Therefore, the answer is 24m^{3}n^{2} – 40m^{2}n^{3} + 32m^{2}n^{2}

**Question 2.**

Find the product of p – q – r and -6p^{2
}**Solution:**

From the given expressions, -6p^{2} is the monomial, and p – q – r is the polynomial.

Write the monomial and the polynomial in the same row and then separate them by using the multiplication sign.

-6p^{2} × (p – q – r)

Apply the distributive law and multiply each term of the polynomial by a monomial.

According to the distributive law, a × (b + c) = (a × b) + (b × c)

-6p^{2} × (p – q – r) = -(6p^{2} × p) + (6p^{2} × q) + (6p^{2} × r)

Multiply the coefficients and variabes separately.

-(6p^{2} × p) + (6p^{2} × q) + (6p^{2} × r) = -(6p^{2 + 1}) + (6p^{2}q) + (6p^{2}r) = -6p^{3} + 6p^{2}q + 6p^{2}r

Therefore, the answer is -6p^{3} + 6p^{2}q + 6p^{2}r

**Question 3.**

Find the product of a^{2} + 2ab + b^{2} + 1 by c

**Solution:**

From the given expressions, c is the monomial and a^{2} + 2ab + b^{2} + 1 is the polynomial.

Write the monomial and the polynomial in the same row and then separate them by using the multiplication sign.

c × (a^{2} + 2ab + b^{2} + 1)

Apply the distributive law and multiply each term of the polynomial by a monomial.

According to the distributive law, a × (b + c) = (a × b) + (b × c)

c × (a^{2} + 2ab + b^{2} + 1) = (c × a^{2}) + (c × 2ab) + (c × b^{2}) + (c × 1)

Multiply the coefficients and variabes separately.

(c × a^{2}) + (c × 2ab) + (c × b^{2}) + (c × 1) = ca^{2} + 2abc + cb^{2 }+ c

Therefore, the answer is ca^{2} + 2abc + cb^{2 }+ c

**Question 4.**

Find the product of 5m^{3} – 13mn + 10n^{2} and -2mn.

**Solution:**

From the given expressions, -2mn is the monomial, and 5m^{3} – 13mn + 10n^{2} is the polynomial.

Write the monomial and the polynomial in the same row and then separate them by using the multiplication sign.

-2mn × (5m^{3} – 13mn + 10n^{2})

Apply the distributive law and multiply each term of the polynomial by a monomial.

According to the distributive law, a × (b + c) = (a × b) + (b × c)

-2mn × (5m^{3} – 13mn + 10n^{2}) = -(2mn × 5m^{3}) + (2mn × 13mn) – (2mn × 10n^{2})

Multiply the coefficients and variabes separately.

-(2mn × 5m^{3}) + (2mn × 13mn) – (2mn × 10n^{2}) = -(10m^{ 1+ 3}n) + (26m^{ 1 + 1}n^{1 + 1}) – (20mn^{1 + 2}) = -10m^{4}n + 26 m^{2}n^{2} – 20mn^{3}

Therefore, the answer is -10m^{4}n + 26 m^{2}n^{2} – 20mn^{3}

### FAQs on Multiplying Polynomial Expressions by Monomials

**1. What is the result while Multiplying a Polynomial by Monomial?**

The resultant is polynomial if you multiply a Polynomial by a Monomial.

**2. Tell me one property you use while doing multiplication of a monomial by a polynomial?**

We use the distributive property while doing multiplication of monomial by a polynomial

**3. What are the steps to multiply a polynomial by a monomial?**

We will take the monomial and polynomial and then multiply each term of the polynomial by a monomial. By using the distributive property, we multiply and simplify the polynomial and get the output.

** 4. Is 2x + 3y a monomial?**

No, 2x + 3y is not a monomial. The monomial will have only one term. x, y, 2xy, are some of the examples of monomial.

** 5. Multiply 2x with 3y + 4z**

2x (3y + 4z) = (2x × 3y) + (2x × 4z) = 6xy + 8xz.

### Summary

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