Multiplying a Monomial by a Polynomial: Steps & Examples

Monomials and polynomials are part of the algebraic expression which is a set of constants and variables related by one or more of the four arithmetic expressions or just expressions, like addition (+), subtraction (-), multiplication (x), and division (÷). The algebraic expression can be classified as monomial, binomial, etc depending upon the number of terms it has. Using monomials and polynomials, we can perform basic operations like addition, subtraction, multiplication, and division. 

Keyterms: Monomials, Polynomials, Addition (+), Subtraction (-), Multiplication (x), Division (÷), Binomial

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What is an Algebraic Expression?

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An algebraic expression is made up of constants and variables linked by any or all of the four arithmetic operations such as addition, subtraction, multiplication, and division. Some examples of algebraic expressions are y + 4, 2x + 5, xy² + 2x + 6, 2xy³ + 3y + 1, etc. We know, an algebraic expression is made of variables and constants. Let's understand what is variable and constant. 

Algebraic Expression

Let us take an algebraic expression- 3x + 4. The expression 3x + 4 is formed from the variable x and constants 3 and 4. The value of a variable is not fixed, we can put any value in the variable. The constant has a fixed value, which means the value of the constant can not be changed. 

The video below explains this:

Polynomials Detailed Video Explanation:

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Classification of Algebraic Expressions

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An algebraic expression can be categorized depending upon the number of terms it contains. We can classify the algebraic expression as monomial, binomial, trinomial, quadrinomial, polynomial. Let us understand it in detail.

• Monomial: An algebraic expression which has only one term is called a monomial. Example- 2x, 5y², 4xy, etc.
• Binomial: An algebraic expression with two terms is called a binomial. Example- 2x+1, 5y²-5, 4xy+2x, etc.
• Trinomial: An algebraic expression which has three terms is called a trinomial. Example- x²+2x+1, 5y²-xy+5, 4xy+2x+1, etc.

Algebraic Expression

• Quadrinomial: An algebraic expression with four terms is called a quadrinomial. Example- x²+2x+y-1, 5y²-xy+x+5, 4x³y+y²+2x+1, etc.
• Polynomial: An algebraic expression with two or more than two terms is called a polynomial. Binomial, trinomial, quadrinomial all can be generally termed as polynomials. Example- x²+2x+2, 6y-3, 5y²-xy+x+5, 4x³y+y², etc.

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Multiplying a Monomial by a Polynomial

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When a monomial is multiplied by a polynomial, the outcome is a new polynomial. When multiplying a monomial by a polynomial, there are a few rules we need to follow. To multiply a monomial by a polynomial, we apply the distributive property. Let's understand this with the help of an example:

Let's take a monomial x and polynomial y+z+1. We will multiply 3x and 4xy+2y+1 using distributive property. 

The distributive property says,

a(b+c) = ab + ac

So similarly, 

x(y+z+1) = x(y) + x(z) + x(1) = xy + xz + x

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Mutilplication of a Constant Monomial & Polynomial

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When we multiply a constant monomial with a polynomial, the monomial is multiplied by the coefficient of the polynomial. Let us understand this with an example. We have a constant monomial 2 and a polynomial 4xy+ 3x+ 3.

Step 1: We will multiply constant monomial 2 to the coefficient of the first term 4xy, i.e. 2 x 4 = 8 and place the variable as it is. We will get 8xy.

Step 2: We will place the appropriate arithmetic operator after the term.

Step 3: Moving further, we will multiply the constant monomial 2 to the coefficient of the second term 3x y. 2 x 3 = 6. We will get 6x.

Step 4: Similarly, we will multiply constant monomial 2 to the third term i.e. 3. We will get 6.

Step 5: Upon multiplying we will get 8xy+6x+6.

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Multiplication of a Variable Monomial and Polynomial with Same Variable

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When we multiply a variable monomial with a polynomial with the same variable, the law of exponents is used for the multiplication. Let us understand this with an example. We have a variable monomial 2x and polynomial 4x²+ 3x.

Step 1: We will multiply variable monomial 2x to the first term 4x². The coefficient of the monomial will multiply with the coefficient of the polynomial and the variable of both the expressions will multiply. i.e. 2 x 4 = 8 and x(x²) =x³. We will get 8x³.

Step 2: We will place the appropriate arithmetic operator after the term.

Step 3: Moving further, we will multiply the multiply variable monomial 2x to the second term 3x i.e. 2 x 3 = 6 and x(x)=x². We will get 6x².

Step 4: Upon multiplying both the expressions, we will get 8x³ + 6x².

Multiplying Monomial with Polynomial

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Multiplication of a Variable Monomial and Polynomial With Different Variable

When we multiply a variable monomial with a polynomial with a different variable, the coefficient of the monomial will multiply with the coefficient of the polynomial and the variable of both the expressions will multiply. Let us understand this with an example. We have a variable monomial 2x and polynomial 4x²y+ 3y+5.

Step 1: We will multiply variable monomial 2x to the first term 4x²y. The coefficient of the monomial will multiply with the coefficient of the polynomial and the variable of both the expressions will multiply. i.e. 2 x 4 = 8 and x(x²y) =x³y. We will get 8x³y.

Step 2: We will place the appropriate arithmetic operator after the term.

Step 3: Moving further, we will multiply the multiply variable monomial 2x to the second term 3y i.e. 2 x 3 = 6 and x(y)=xy. We will get 6xy.

Step 4: Similarly, we will multiply 2x to the third term i.e. 5. We will get 10x.

Step 5: Upon multiplying both the expressions, we will get 8x³y + 6xy+10x.

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Things to Remember

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• The algebraic expression can be classified as monomial, binomial, etc depending upon the number of terms it has. 
• When a monomial is multiplied by a polynomial, the outcome is a new polynomial. When multiplying a monomial by a polynomial, we apply the distributive property.
• When we multiply a constant monomial with a polynomial, the monomial is multiplied by the coefficient of the polynomial.
• When we multiply a variable monomial with a polynomial with the same variable, the law of exponents is used for the multiplication. The coefficient of the monomial will multiply with the coefficient of the polynomial and the variable of both the expressions will multiply. 
• When we multiply a variable monomial with a polynomial with a different variable, the coefficient of the monomial will multiply with the coefficient of the polynomial and the variable of both the expressions will multiply.