Multiplying a Monomial by a Polynomial: Algebraic expressions can be classified into some types based on the number of terms, viz monomial, binomial, trinomial, quadrinomial, etc. Monomial is an algebraic expression with only one term, binomial with two terms, trinomial with three terms.

A polynomial is an algebraic expression with one or more than one term. So, binomials and trinomials are also polynomials. We can perform fundamental operations such as addition, subtraction, multiplication, and division using monomials and polynomials. The multiplication of a monomial and a polynomial follow some specific rules. This article will tell us about monomials, polynomials, types, and the process of multiplying a monomial by a polynomial.

Algebraic Expression

The combination of constants and variables connected by any or all of the four arithmetic operations such as addition \((+),\) subtraction \((-),\) multiplication \((×),\) and division \((÷)\) is known as an algebraic expression. For example, \(x + 7,a – 5,4y + 5,3y,6{a^2}b\) etc.

Types of Algebraic Expressions

We can classify the algebraic expressions based on their number of terms. These are,

Monomial

It is an algebraic expression that includes only one term.

Example: \(5x, – 6xy,3{a^2}c,9, – 10\) etc., are monomials.

Binomials

An algebraic expression including two unlike terms is called a binomial.

Example: \((2x + 3y),(6 – 3x),\left( {{y^2} – x{y^2}} \right)\) etc., are binomials.

Trinomials

An algebraic expression containing three unlike terms is called a trinomial.

Example: \((a + y + c),(x + 2a + 6z),\left( {{a^3} – {y^2} – {z^3}} \right)\) etc., are trinomials.

Polynomials

The algebraic expressions with the exponents of the variables as whole numbers are polynomials.
Polynomials are algebraic expressions involving variables and constants with whole-number exponents of the variables.

Thus, monomials, binomials, trinomials, quadrinomials, etc., are polynomials.

Multiplication of a Monomial by a Polynomial

The multiplication of a monomial by a polynomial gives a new polynomial as a result. There are some definite rules of multiplication of a monomial by a polynomial. Let us discuss the methods of multiplication.

We use the distributive property to multiply a monomial with a polynomial. Let’s say monomial \(a\) has to be multiplied with trinomial \((b+c+d).\) By using the distributive property, the above result can be written as, \(a(b + c + d) = ab + ac + ad\)

Multiplying a Constant Monomial with a Polynomial

• Step 1: We are considering a monomial \(10\) and a polynomial \(2a+3b.\) In this case, \(10\) is a constant monomial, and \(2a+3b\) is a polynomial with two terms. We will multiply the constant monomial with the coefficient of the first term of the polynomial from the left. It gives \(10×2=20.\) After that, we will write the variable \((a)\) after \(20.\) Hence, we get \(20a,\) after multiplying the monomial with the first term of the polynomial.
• Step 2: Next, we will write the arithmetic operator \((+)\) present in the polynomial between two terms.
• Step 3: We need to follow the same rule as step 1 with the second term of the polynomial, and we get, \(30b\)
• Hence, the answer is \(20a+30b.\)

Multiplying a Variable Monomial with a Polynomial of Same Variables

Considering one variable monomial and one polynomial with \(2a,\left( {4a + 3{a^2} + {a^3}} \right)\)

The monomial and polynomial are containing the same variable. In this case, we need to apply the law of exponents in each step.

Step 1: We are considering a monomial \(2a\) and a polynomial \({4a + 3{a^2} + {a^3}}.\) In this case, \(2a\) is a variable monomial, and \(\left( {4a + 3{a^2} + {a^3}} \right)\) is a polynomial with three terms. We will multiply the monomial by each term of the polynomial using the distributive property. We need to remember; we will multiply the coefficient of the monomial by the coefficient of each term of polynomial similarly, variable with variable.

The first term of the polynomial is \(4a.\) Now, \(4a \times 2a = 8{a^2}.\) (using the law of the exponents)

Step 2: Similarly, the second term of the polynomial is \(3{a^2}.\) Now multiplying \(2a\) by \(3{a^2}.\) we have, \(6{a^3}\)

Step 3: Similarly, the third term of the polynomial is \({a^3}.\) Now multiplying \(2a\) by \({a^3}\) we have, \(2{a^4}\)

Hence, the answer is  \(2a \times \left( {4a + 3{a^2} + {a^3}} \right) = 8{a^2} + 6{a^3} + 2{a^4}.\)

Multiplying a Variable Monomial with a Polynomial of Different Variables

Considering one variable monomial \(4{x^2}\) and one polynomial \((2x+y+z)\)

• Step 1: We are considering a monomial \(4{x^2}\) and a polynomial \(2x+y+z.\) In this case, \(4{x^2}\) is a variable monomial, and \(2x+y+z\) is a polynomial with three terms. We will multiply the monomial by each term of the polynomial using the distributive property.
• The first term of the polynomial is \(2x.\) Now, \(4{x^2} \times 2x = 8{x^3}\) (using the law of the exponents)
• Step 2: The second term of the polynomial is \(y.\) Now multiplying \(4{x^2}\) by \(y\) we have, \(4{x^2}y\) 
• Step 3: The third term of the polynomial is \(z.\) Now multiplying \(4{x^2}\) by \(z\) we have, \(4{x^2}z\)

Hence, the answer \(4{x^2}(2x + y + z) = 8{x^3} + 4{x^2}y + 4{x^2}z.\)

Solved Examples on Multiplying a Monomial by a Polynomial

Q.1. Multiply \(x\) and \({x^2} + x.\)
Ans: Given, \(x\) and \({x^2} + x\)
Step 1: We are considering a monomial \(x\) and a polynomial \({x^2} + x.\) In this case, \(x\) is a variable monomial and \({x^2} + x\) is a polynomial with two terms. We will multiply the monomial by each term of the polynomial using the distributive property. We need to remember; we will multiply the coefficient of the monomial by the coefficient of each term of polynomial and variable with variable.
The first term of the polynomial is \({x^2}.\) Now, \(x \times {x^2} = {x^3}\) (using the law of the exponents)
Step 2: Similarly, the second term of the polynomial is \(x.\) Now multiplying \(x\) by \(x\) we have, \({x^2}\)
Hence, the answer is \(x\left( {{x^2} + x} \right) = {x^3} + {x^2}\)

Q.2. Multiply \(5\) and \(4{x^3} + 2y\)
Ans: We are considering a monomial \(5\) and a polynomial \(4{x^3} + 2y.\) In this case, \(5\) is a constant monomial, and \(4{x^3} + 2y\) is a polynomial with two terms.
Step 1: We will multiply the constant monomial with the coefficient of the first term of the polynomial from the left. It gives \(5×4=20.\) After that, we will write the variable \(\left( {{x^3}} \right)\) after \(20.\) Hence, we get \(20{x^3}\) after multiplying the monomial with the first term of the polynomial.
Step 2: Next, we will write the arithmetic operator \((-)\) present in the polynomial between two terms.
Step 3: We need to follow the same rule as step 1 with the second term of the polynomial, and we get, \(10y\)
Hence, the answer is \(20{x^3} + 10y\)

Q.3. Multiply \(3b\) and \({b^2} + 2{b^4} + b\)
Ans: Step 1: We are considering a monomial \(3b\) and a polynomial \({b^2} + 2{b^4} + b.\) In this case, \(3b\) is a variable monomial, and \({b^2} + 2{b^4} + b\) is a polynomial with three terms. We will multiply the monomial by each term of the polynomial using the distributive property.
The first term of the polynomial is \({b^2}.\) Now, \(3b \times {b^2} = 3{b^3}\) (using the law of the exponents)
Step 2: Similarly, the second term of the polynomial is \(2{b^4}.\) Now multiplying \(3b\) by \(2{b^4}\) we have, \(6{b^5}\)
Step 3: Similarly, the third term of the polynomial is \(b.\) Now multiplying \(3b\) by \(b\) we have, \(3{b^2}\)
Hence, the answer is \(3b \times \left( {{b^2} + 2{b^4} + b} \right) = 3{b^3} + 6{b^5} + 3{b^2}.\)

Q.4. Multiply \(2abc\) and \({a^2}b + 6\)
Ans: Given, \(2abc,{a^2}b + 6\)
We will multiply the coefficients. We get \(2×1=2\) for the first term, and then we will multiply the variables using the exponent rule wherever required. Here, the variable parts are \(abc,{a^2}b.\) Multiplying these, we get, \(abc \times {a^2}b = \left( {a \times {a^2}} \right) \times (b \times b) \times c = {a^3}{b^2}c\) (as we added the exponents of the same variables as per the exponent rule.)
Next, we will multiply 2abc by the second term of the polynomial, which is \(6.\) It gives \(12abc\)
Hence, the answer is \({a^3}{b^2}c + 12abc\)

Q.5. Multiply \(5{x^2}\) and \((3x + y + z)\)
Ans: Step 1: We are considering a monomial \(5{x^2}\) and a polynomial \(3x + y + z.\) In this case, \(5{x^2}\) is a variable monomial, and \(3x + y + z\) is a polynomial with three terms. We will multiply the monomial by each term of the polynomial using the distributive property.
The first term of the polynomial is \(3x.\) Now, \(5{x^2} \times 3x = 15{x^3}\) (using the law of the exponents)
Step 2: The second term of the polynomial is y. Now multiplying \(5{x^2}\) by y we have, \(5{x^2}y\)
Step 3: The third term of the polynomial is \(z.\) Now multiplying \(5{x^2}\) by \(z\) we have, \(5{x^2}z\)
Hence, the answer \(5{x^2}(3x + y + z)\)
\( = 15{x^3} + 5{x^2}y + 5{x^2}z.\)

Summary

In this article, we learnt about monomial, polynomials, types of polynomials, multiplying one monomial by a polynomial. We also discussed the method of multiplication of a monomial with different types of polynomials. At last, we solved some examples of the multiplication of a monomial by a polynomial.

FAQs

Q.1 How do you multiply a monomial by a polynomial?
Ans: The constant coefficient of a monomial is multiplied by the constant coefficient of each term of the polynomial. The variable part of the monomial is multiplied with the variable part of each term of the polynomial. We use the exponent rule during multiplication wherever it is needed.

Q.2. How do you multiply a polynomial by a polynomial?
Ans: We use the distributive property to multiply a polynomial with a polynomial. By using the distributive property, the result can be written as, \((a + b)(c + d + e) = ac + ad + ae + bc + bd + be.\)

Q.3. How do you multiply a monomial by another monomial?
Ans: First, we will multiply the coefficients. Next, we will multiply the variables using the rule of the exponent wherever it is required.

Q.4. What are variable monomials?
Ans: It is a monomial with a variable or the variables, or we can represent it by letters such as \(a, b, c, x, y, z,\) etc.
Variable monomials may hold only or more variables. \(2a,6{x^2}, – 7xy,abc,\) etc., are the variable monomials.

Q.5. What do you understand by a binomial and a trinomial?
Ans: A polynomial including two unlike terms is called a binomial.
Example: \((2x + 3y),(6 – 3x),\left( {{y^2} – x{y^2}} \right),\) etc.
A polynomial containing three unlike terms is called a trinomial.
Example: \((a + y + c),(x + 2a + 6z),\left( {{a^3} – {y^2} – {z^3}} \right),\) etc.

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