Multiplying monomials means two multiply two single terms. The single term may have a coefficient and exponents. Multiplying coefficients and then variables gives the complete result.

## Monomials:

A Monomial is a single term having coefficients, variables, or a product of coefficients and variables.

It can be positive or negative.

Examples are 5x, 45, x

## Multiplication of a monomial by monomial

While doing multiplication you need to follow two steps:

### Step 1: Multiply coefficients

Multiply the coefficient of the first term by the coefficient of the second term

### Step 2: Multiply exponents

Multiply variables of the first term with second term variables

**Case 1:** If there are similar variables add the power.

Example: **y x y**

Here the power of y is 1 in both cases. Adding power 1 + 1 = 2

So y x y = y^{2}

**Case 2**: If the variables are not alike then do normal multiplication.

Example: y x z = yz

## Example of multiplication of monomials

Example | Solution |

2^{4} ⋅ 2^{5} | 2^{4+5}=2 ^{9} |

y^{4} ⋅ y^{5} | y^{4+5}=y ^{9} |

5^{2} ⋅ y^{3} | 25y^{3}Note: If you keep the solution same as question here, it works are first term do not contain variable and second term lacks coefficient. |

3y ⋅ 5y | = (3 x 5) ⋅ (y x y) = 15y ^{2} |

3y ⋅ 4z | = (3 x 4) ⋅ (y x z) =12yz |

## Multiplying Monomials and a Binomial

To multiply a monomial by a binomial, simply use the distributive property.

To do the multiplication of monomial and binomial (two terms) follow two steps:

### Step 1: Multiply monomial with first term

Multiply the monomial with the first term of the binomial expression.

Multiplying with the first becomes similar to multiplying two monomials. Now you can just repeat the process above

### Step 2: Multiply monomial with second term

Multiply the monomial with the second term of the binomial which becomes similar to multiplying monomials.

Once done check if the result can be further reduced.

## Examples of multiplication of monomial with binomial

Example | Solution |

3 ⋅ (2y + 3) | = (3 ⋅ 2y) + (3 ⋅ 3) = 6y + 9 Here first term is product of coefficient and variable. Second term is just coefficient. Terms are not alike so addition further cannot be done. |

2 ⋅ (2y + 3z) | = (2 ⋅ 2y) + (2 ⋅ 3z) =4y + 6z Further reduction cannot be done as terms are different |

3y ⋅ (3 – 4z) | = (3y ⋅ 3) – (3y ⋅ 4z) = 9y – 12yz |

3y ⋅ (4y – 5z) | = (3y ⋅ 4y) – (3y ⋅ 5z) = 12y ^{2} – 15yz |

## Multiplying Monomials with Trinomial:

The distributive property can also be applied when computing the product of a monomial and a trinomial or any other polynomial. Consider this example,

** 9n(6n^2 – 7n + 10)** or equivalently, **(9n) (6n ^{2} – 7n + 10)**

Using the distributive property and multiplying 9n by each of the three other terms, we have

9n(6n

^{2}) – 9n(7n) + 9n(10) = 54n

^{3}– 63n

^{2}+ 90n

In case you have a negative monomial, always remember to check the sign of the product of every term. Let’s consider this product:

-10x

^{2}(18x

^{2}– 5x + 7)

Using the distributive property, we have

-10x

^{2}(18x

^{2}) + 10x

^{2}(5x) – 10x

^{2}(7)

After performing the indicated multiplication associated with each term, this simplifies to

-180x

^{4}+ 50x

^{3}– 70x

^{2}

## Conclusion:

- The product of powers property can be used when doing the multiplication of exponents
- Always multiply coefficients and then variables by using like and different terms

## How do you multiply monomial with different bases?

Multiply coefficients and keep the variables as it is.

Example: 2y x 6z = (2 x 6)(y x z)

= 12yz

## How do multiply 4 monomials?

Follow the same two steps mentioned above in the article.

Multiply all coefficients and then multiply all variables

(4z)(5y)(3z^{2})(4y)

Step 1: 4 x 5 x 3 x 4 = 240

step 2: z x y x z^2 x y = (z x z^{2})(y x y) = y^{2}z^{3}

Result: 240y^{2}z^{3}

## How to multiply monomials with polynomials

polynomials means more than one term which can be binomial, trinomials and so on.

Follow the above two steps to do the multiplication

## Multiplying monomials worksheet

Click on the download button to download multiplying monomials worksheet

## What is the best method in multiplying monomials?

The best approach is to first multiply all coefficients and then find alike exponents and multiply them

(4z)(5y)(3z^{2})(4y)

Step 1: 4 x 5 x 3 x 4 = 240

step 2: z x y x z^2 x y = (z x z^{2})(y x y) = y^{2}z^{3}

Result: 240y^{2}z^{3}

## How do you solve monomials?

The best approach is to first multiply all coefficients and then find alike exponents and multiply them

(4z)(5y)(3z^{2})(4y)

Step 1: 4 x 5 x 3 x 4 = 240

step 2: z x y x z^2 x y = (z x z^{2})(y x y) = y^{2}z^{3}

Result: 240y^{2}z^{3}

## Is there multiplying monomials quiz?

yes. Below is a Quizlet that will help you test yourself

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