Multiplying Polynomials
Multiplication is one of the arithmetic operations which can be applied to polynomials. Multiplying polynomials is one of the simplest things in algebra. Polynomials can be easily multiplied by using their rules. When multiplying polynomials we multiply coefficients together and variables together. In this chapter, we will discuss the multiplication of polynomials, their rules, and the steps to multiply polynomials.
1.  Multiplication of Polynomials 
2.  How to Multiply Monomials? 
3.  How to Multiply Binomials? 
4.  How to Multiply a Monomial With a Binomial? 
5.  FAQs on Multiplying Polynomials 
Multiplication of Polynomials
Polynomial multiplication is a method for multiplying two or more polynomials together. The terms of the 1^{st }polynomial are multiplied with the 2^{nd} polynomial to get the resultant polynomial. Based on the types of polynomials we use, there are different ways of multiplying them. The rules for the multiplication of polynomials are different for each type of polynomial. To multiply polynomials, the coefficient is multiplied with a coefficient, and the variable is multiplied with a variable.
Multiplying Polynomials with Exponents
When the polynomials are multiplied it is possible they can be monomial, binomial, or trinomial. In order to multiply any two polynomials the steps used are:
 Multiply the coefficients
 Multiply the variables using exponent rules as per the requirement.
Let us understand how to multiply polynomials with exponents using an example.
Example: Multiply 2x^{3} with 3x^{2}
We will follow the same procedure for multiplying polynomials with exponents as we had done above.
 Step 1: First we will multiply the coefficients i.e., 2 × 3 = 6
 Step 2: Next, we will multiply the variables but in this case, the powers of both the variables will be added as per the rules of exponents i.e., x^{3} × x^{2} = x^{5}
The final answer is 2x^{3} × 3x^{2} = 6x^{5}
Multiplying Polynomials with Different Variables
It is possible to multiply polynomials with different variables too. The steps to multiply polynomials with different variables are:
 Multiply the coefficients
 Multiply the variables and use rules of exponents wherever necessary.
Example: Multiply 5x^{2} with 3y.
 Step 1: We will first multiply the coefficients of both the polynomials i.e., 5 × 3= 15
 Step 2: Since the above polynomials have two different variables, they cannot be multiplied. Hence, we will keep them the same.
The final answer is 5x^{2}× 3y = 15x^{2}y
How to Multiply Monomials?
Monomials are polynomials having just one term, consisting of a variable and its coefficient. Hence the steps to determine the product of two or three monomials follow the same steps as we learned above. It is possible to multiply monomials more than three too using the same steps we will learn for the below examples.
Multiplication of Two Monomials
When multiplying monomials, we need to follow certain rules similar to multiplying polynomials. Let us understand by taking two monomials, 3x and 2x.
 Step 1: In the above monomials, the common variable is x. We will multiply the variable with the variable. Hence, we get x × x = x^{2}.
 Step 2: In the next step, we will multiply the coefficients of both the monomials to get 2 × 3 = 6. Thus, multiplying the polynomials 2x and 3x gives 6x^{2} as the result.
Multiplication of Three Monomials
To multiply three monomials, we will use the same method as that used for multiplying two monomials. Let us understand the method with an example.
Example: Multiply 2x, 3y, and 6z.
 Step 1: First we will multiply the variables together i.e., x × y × z = xyz
 Step 2: Next we will multiply the coefficients of all the three terms i.e., 2 × 3 × 6 = 36
Thus, the multiplication result can be shown as 2x × 3y × 6z = 2 × 3 × 6 × x × y × z = 36xyz
How to Multiply Binomials?
Binomials are a particular kind of polynomials consisting of only two terms. They can be multiplied in two ways:
 Distributive Property
 Box Method
Multiplying Binomials by Distributive Property
For multiplying binomials, we use the distributive property. Let's multiply a binomial (a+b) with another binomial (c+d).
 Step 1: Write both the binomials together i.e., (a + b)(c + d)
 Step 2: Out of the two brackets, keep one bracket constant, let's say (c + d).
 Step 3: Now multiply each and every term from the other bracket i.e., (a + b) with (c + d).
Example:
Multiply (2x+3)(4x+5)
The above polynomials can be solved as:
(2x + 3)(4x + 5) = 2x(4x + 5) + 3(4x + 5)
⇒ 8x^{2} + 10x + 12x + 15
⇒ 8x^{2} + 22x + 15
Multiplying Binomials by Box Method
Two binomials can also be multiplied using the box method. The terms are written across a box and their corresponding products are written inside the box.
Example:
Multiply (x + 7) with (x + 3)
Solution: Let's write the polynomials (x + 7) horizontally and (x + 3) vertically. Take the sign with its corresponding term on the right. After multiplying the corresponding terms, we get:
Thus, the above multiplication method is known as the box multiplication of two binomials. We now have (x^{2} + 7x + 3x + 21) as the sum. Thus, the final product will be (x^{2} + 10x + 21).
How to Multiply a Monomial with a Binomial?
As we did above, to multiply a monomial with a binomial, we have to use the distributive property. Let's say monomial a has to be multiplied with binomial (b + c). By distributive property, the above product can be written as: a(b + c) = ab + ac.
Example
Multiply 3y with (5x + 2z)
Solution:
3y(5x + 2z) = 3y × 5x + 3y × 2z
⇒ (3 × 5 × y × x) + (3 × 2 × y × z) = 15yx + 6yz
Topics Related to Multiplying Polynomials
Check out these interesting articles to learn more about multiplying polynomial and its related topics.
 Multiplying Polynomials Calculator
 Monomial
 Trinomial
 Multiplying Binomials Calculator
 Polynomial Calculator
Tips to Remember
 When multiplying polynomials, the coefficient will be multiplied with a coefficient and the variable will be multiplied with a variable.
 Polynomials can also be solved using the distributive property, box method, or grid method.
 When multiplying polynomials with exponents, the rules of exponents have to be used.
Multiplying Polynomials Examples

Example 1: Simplify xz(x^{2} + z^{2}) by using rules of multiplication of polynomials.
Solution: xz(x^{2} + z^{2}) = (xz × x^{2}) + (xz × z^{2})
⇒ x^{3}z + xz^{3}
Therefore, the product is x^{3}z + xz^{3}

Example 2: Find the product: (2x + 3y)(4x  5y)
Solution: By using distributive property for multiplying polynomials, we get
2x(4x 5y) + 3y(4x  5y) = 8x^{2}  10xy + 12xy 15y^{2}
⇒ 8x^{2} + 2xy  15y^{2}
Therefore, the product is 8x^{2} + 2xy  15y^{2}

Example 3: A cuboid has sides measuring 2y, 3x and 5z as its length, breadth, and height respectively. Find the volume of the cuboid.
Solution: As we know, the volume of cuboid = length × breadth × height. Here, all the side lengths are given in the form of monomials, so by applying rules for multiplying monomials, we get,
⇒ Volume = 2y × 3x × 5z = 30xyz
Therefore, the volume of the cuboid is 30xyz cubic units.
FAQs on Multiplying Polynomials
How do you Multiply Three Polynomials?
Multiplication of three polynomials is a twostep process that involves the following two steps:
 Multiplication of coefficients
 Multiplication of the variables using Laws of Exponents as and when required.
Let's take an example to understand the multiplication of three polynomials.
Example: Multiply (3m+2), 4n^{2}, and 7p.
 The above given three polynomials are written as (3m+2)× 4n^{2}× 7p
 By using distributive property of polynomial multiplication we get, ((3m× 4n^{2})+(2× 4n^{2}))× 7p = (12mn^{2} + 8n^{2})7p = 84mn^{2}p + 56n^{2}p
Thus, the above multiplication can be shown as (3m+2)× 4n^{2}× 7p = 84mn^{2}p + 56n^{2}p.
How can we Multiply Polynomials Using the Box Method?
Two or more polynomials can be multiplied using the box method. The terms are written across a box and their corresponding products are written within the box.
Example: (3x^{2}+ 2x + 4)(4x + 5)
3x^{2}+2x+4 will be written on the vertical side of the box while 4x+5 will be written on the horizontal side of the box, or viceversa. Then, first, we will multiply 3x^{2} by 4x, then 3x^{2} by 5, and write the products in the corresponding section of the box. Secondly, we will multiply 2x by 4x and 2x by 5 and write down the products. The final column of the box is filled by multiplying 4 by 4x and 4 by 5. At last, we will add all six terms obtained to get the final answer.
Therefore, the result of the multiplication of both the polynomials is (12x^{3}+23x^{2}+26x+20).
How do you Multiply Binomials Using the Grid Method?
The steps to multiply polynomials by a box method or the grid method is as follows:
Example: (x + 6)(2x + 3)
x+6 will be written on the vertical side of the box while 2x+3 will be written on the horizontal side of the box, or viceversa. Multiply each term with the respective terms. Therefore, the product which we get is (2x^{2}+ 15x + 18).
How Many Methods are there for Multiplying Polynomials?
There are two methods for multiplying polynomials:
 Distributive Property
 Box Method
What does FOIL Stand for in Multiplying Binomials?
FOIL stands for First, Outer, Inner Last in multiplying binomials. The binomials are multiplied as:
 Step 1: Multiply the first term of each binomial.
 Step 2: Now multiply the outer term of each binomial.
 Step 3: Once this is done, now multiply the inner terms of the binomials.
 Step 4: Now the last terms are multiplied.
 Step 5: Once all the above four steps are done, the products obtained as each step are added, like terms are combined and the answer is simplified.
What is the Best Method for Multiplying Polynomials?
The best method for multiplying polynomials is the distributive property of multiplying polynomials. The steps to multiply a polynomial using the distributive property are:
 Step 1: Write both the polynomials together.
 Step 2: Out of the two brackets, keep one bracket constant.
 Step 3: Now multiply each and every term from the other bracket.
How Do You Multiply Two Trinomials Together?
Two trinomials can be multiplied together by using the box method as well as the distributive property. Let's take an example to understand the multiplication of two trinomials.
Example: Multiply (5xy + 2x + 3) with (x^{2}+ 3xy + 7)
 The above given two trinomials are written as (5xy + 2x + 3)× (x^{2}+ 3xy + 7)
 By using distributive property of polynomial multiplication we get, (5xy + 2x + 3)× (x^{2}+ 3xy + 7) = 5x^{3}y + 15x^{2}y^{2} + 2x^{3} + 6x^{2}y + 44xy+ 3x^{2} + 14x + 21
Thus, the above multiplication can be shown as (5xy + 2x + 3)× (x^{2}+ 3xy + 7) = 5x^{3}y + 15x^{2}y^{2} + 2x^{3} + 6x^{2}y + 44xy + 3x^{2} + 14x + 21.
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