Multiplying a monomial by a monomial

Introduction

Multiplication is one of the basic mathematical operations used in algebraic expressions. We can classify algebraic expressions according to the number of terms they contain, such as monomial, binomial, trinomial, quadrinomial, or polynomial. A monomial expression is a one-term algebraic expression that contains a variable and its coefficients. A monomial multiplied by a monomial: When we multiply a monomial by a monomial, the resulting product will also be a monomial. For example, x, y, 2x, 2y, x2, y2, etc. are all monomials. Monomials cannot have negative exponents.

Now, if we multiply the monomial by the monomial, the result is the monomial. The coefficients of the monomial are multiplied, and then the variables are multiplied. For example, the product of two monomials such as 2x and 2y equals 4xy. If two monomials have the same variable and the same exponent, then we need to use the law of exponentials.

Monomial

Monomials are a type of polynomials with only one term. Monomials algebraic expressions are a type of expression that have only a single term, but can also have multiple variables and higher degrees. For example, \(9{x^3}yz\) is the monomial, where 9 is the coefficient, x, y, z are the variables, and 3 is the degree of the monomial. Similar to polynomials, we can perform different operations on monomials, such as addition, subtraction, multiplication and division.

Monomial example

Let’s consider some variables and  monomial examples:

\(p\) – a variable with a degree of one.

\(5{p^2}\) – The factor is 5 and the degree is 2.

\({p^3}q\) – has two variables (p and q) with degree \(4{\rm{ }}(i.e.,{\rm{ }}3 + 1).\)

\( – 6ty{\rm{ }}–{\rm{ }}t\) and \(y\) are two variables with a coefficient of\( – 6\) .

Let’s consider \({x^3} + 3{x^2} + 4x + 12\) as a polynomial, where \({x^3},{\rm{ }}3{x^2},{\rm{ }}4x\) and 12 are called monomials.

Parts of a Monomial expression

These are the different parts present in a monomial expression are:

• Variable: The letter that appears in the monomial expression.
• Coefficient: The number to multiply by the variable in the expression.
• Degrees: The sum of the exponents present in the expression.
• literal part: the letters that appear with the exponent value in the expression.

Multiplying Monomials

Monomial multiplication is a method of multiplying two or more monomials at a time. Multiplying a monomial by another monomial yields a monomial as the product. Depending on the type of polynomial we use, there are different ways of multiplying.

There are specific multiplication rules for different types of monomials. The constant factor is multiplied by the constant factor, and the variable is multiplied by the variable.

Multiplying a Constant Monomial With a Variable Monomial

Let us consider two monomials \(7\) and \(6y\) . In this case, \(7\) is a constant monomial, and \(6y\)  is a variable monomial. We multiply the coefficients of the constant monomial with the variable monomial. It gives \(7 \times 6 = 42\) . After that, we write the variable \(\left( y \right)\) after \(42\)

Hence, the answer is
\(7 \times 6y = 42y\) .

Multiplying Two Monomials With Different Variables

Consider two monomials with different variables, \(2{x^3}{\rm{ }}\& {\rm{ }}5y\)

First, we’ll multiply by the coefficients. The coefficient of \(2{x^3}\)  is \(2\) , and the coefficient of \(5y\)  is \(5\) . After multiplying, you get \(2 \times 5 = 10\)

Next, we’ll multiply the variables using the exponential rule wherever needed. Here, the variable part is \({x^3}\) &            \(y\) . Multiplying these together, we get \({x^3} \times y = {x^3}y\) because the variables are different. We can multiply them without using the exponential rule.

Hence, the answer is
\(2{x^3} \times 5y = 10{x^3}y\) .

Multiplying Two Monomials With Same Variable

Let us learn the following steps using the example given below.

Considering two monomials \(4{a^2}\;\& \;3{a^4}\).

First, we will multiply the coefficients. The coefficient of \(4{a^2}\)  is \(4\)  and the coefficient of \(3{a^4}\)  is \(3\) . After multiplying, we get \(3 \times 4 = 12\) .

Next, we will multiply the variables using the rule of the exponents. Here, the variable parts are \({a^2}\)  & \({a^4}\) . Multiplying these we get, \({a^2} \times {a^4} = {a^6}\)  as we added the exponents of the variable as per the rule of the exponent.

Hence, the answer is
\(4{a^2} \times 3{a^4} = 12{a^6}\)

Interesting facts

• Multiplying two monomials will also yield a monomial.
• The sum or difference of two monomials may not result in a monomial.
• An expression with a single term with a negative exponent cannot be treated as a monomial. (i.e,) a monomial cannot have variables with negative exponents.

Solved examples

1. Multiply \(x\)  and \({x^2}\) .

Sol: Given two monomials are \(x\)  & \({x^2}\)

First, we’ll multiply by the coefficients. Both monomials have coefficients of \(1\). Therefore, the product is \(1\).

Next, we’ll use the exponential rule to multiply the variables. Here, the variable part is \(x\) & \({x^2}\). Multiplying these together, we get \(x \times {x^2} = {x^3}\)  because we added the exponent of the variable according to the rule of exponent \(3\)  .

Therefore, the answer is \(x \times {x^2} = {x^3}\).

2. Multiply by \(3x\) and \(4y\) .

Sol: Given two monomials are \(3x{\rm{ }}\& {\rm{ }}4y\)

First, we multiply the coefficients. The coefficient of \(3x\) is \(3\), and the coefficient of \(4y\) is 4. After multiplying, you get \(3 \times 4 = 12\)

Next, we will multiply the variables using the exponential rule wherever needed. Here, the variable parts are \(x,y\) . Multiplying these together, we get \(x \times y = xy\) . Since the variables are different, we can multiply them without using the exponential rule.

Therefore, the answer is \(3x \times 4y = 12xy\) .

3. Multiply by \(7{z^3}\) and \(9{z^2}\)

Sol: Given two monomials are \(7{z^3}{\rm{ }}\& {\rm{ }}9{z^2}\) .

First, we multiply the coefficients. The coefficient of \(7{z^3}\) is 7, and the coefficient of \(9{z^2}\) is 9. After multiplying, we get \(7 \times 9 = 63\) .

Next, we will use the exponential rule to multiply the variables. Here, the variable parts are \({z^3},{\rm{ }}{z^2}\) . Multiplying these together, we get \({z^3} \times {z^2} = {z^5}\) because we added the exponent of the variable according to the exponent rule.

Therefore, the answer is \(63{z^5}\) .

Conclusion

Monomial multiplication is a method of multiplying two or more monomials at a time. Multiplying a monomial by another monomial yields a monomial as the product. Depending on the type of polynomial we use.

Practice questions

1. Find the factorization of the monomial \(10{y^3}\) .

Ans: \(2 \times 5 \times y \times y \times y\) .

2. Multiply \(2abc\)and \({a^2}b\).

Answer: \(2{a^3}{b^2}c\) .

3. Multiply \(8\) and \(6{y^3}\) .

Answer: \(48{y^3}\)

Frequently Asked Questions

1. How do you find the product of two monomials?

Ans: The constant coefficient of one monomial is multiplied by the constant coefficient of another monomial, and the variable is multiplied by one variable.

2. What are the rules in multiplying monomials?

Ans: We will multiply by the coefficient. Next, we’ll multiply the variables using the exponential rule wherever needed.

3. What is a monomial ?

Ans: Monomial   is an algebraic expression that contains only one term.