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Multiplying Polynomials

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Multiplication of Polynomials is multiplying various polynomials and finding their product. Multiplication is the fundamental operation and it can be performed on polynomials, Multiplication of polynomials is achieved by multiplying the coefficients of polynomials with each other and variables with each other. In the multiplication of polynomials, the same variables get multiplied and their power gets added and if other variables are multiplied they are simply written together.

In this article, we will learn about, the multiplication of polynomials, multiplication of monomials, multiplication of binomials, and others in detail.

Multiplication of Polynomials

Polynomials are mathematical expressions made up of constants and variables multiplied together as one. In Polynomial Multiplication we multiply the variables and coefficient of the two polynomials. Suppose we have to find the product of two polynomials, then we multiply the coefficient of the first polynomial with the coefficient of the second polynomial and then we multiply the same variables and change their exponents, then other variables are written as such. A polynomial can be a monomial, binomial, etc., and multiplying each of them is somewhat similar but multiplication of each has its own methods, so multiplication of each is explained in the detail below.

Multiplying Polynomials with Exponents

To multiply the polynomial in which we have the same variables, we multiply the polynomials using the exponents rules. Suppose we are given to multiply the polynomials with the same variables and different exponents then we follow the following steps,

Step 1: Multiply the coefficients of both the variables.

Step 2: To multiply the variables we use the laws of exponents.

For example, Multiply the polynomials 3x5 and 5x2

= (3x5)(5x2)

= (3.5)(x5.x2)

= 15(x5+2)

= 15x7

Multiplying Polynomials with Different Variables

Polynomials with different variables are multiplied together by following the steps that are discussed below,

Step 1: Multiply the coeffieciet of both the variables.

Step 2: Use the laws of exponents to multiply the polynomial or write different variables together to get the required product of the variable.

For example, Multiply the polynomials 3x5 and 5y2

= (3x5)(5y2)

= (3.5)(x5.y2)

= 15x5y2

Multiplying Polynomials

We can multiply polynomial, but we have different types of polynomials, monomials, binomials, etc. Multiplication of monomial with monomial is different and multiplication of binomial with monomila is different, etc. Now lets’s the learn the same in detail.

Monomial Multiplied by a Monomial

Two monomials are easily multiplied. We should follow the following steps to multiply to monomial.

Step 1: Multiply the coefficient of both polynomial together to get the coefficient of resultant polynomial.

Step 2: Multiply the variables of both the polynomial to get required product.

For example, Multiply the polynomials 20x5y and 3xy2

= (20x5y)(3xy2)

= (20.3)(x5y.xy2)

= 60(x5+1)(y1+2)

= 60x6y3

Bionomial Multiplied by a Bionomial

Two bionomials can be multiplied using the distributive properties. The distribuitve properties of the algebra is,

(a + b).(c + d) = a.c + a.d + d.c + d.d

Using this property we can easily multiply two biomomial. To multiply the same follow the steps added below,

Step 1: Write the polynomials in the form (a ± b).(c ± b)

Step 2: Use the property discussed above to open the bracket.

Step 3: Simplify to get the required product.

Example: Multiply (3x + 4y)(5x2 + 2xy)

Solution:

= (3x + 4y)(5x2 + 2xy)

= (3x)(5x2) + (3x)(2xy) + (4y)(5x2) + (4y)(2xy)

= 15x3 + 6x2y + 20x2y + 8xy2

= 15x3 + 26x2y + 8xy2

Monomial Multiplied by a Polynomial

To multiply a polynomial and a monomial we need to multiply each and every term of the polynomial with monomial.

Examples: Find the product of 5x and (5x2 + 2x + 6)

Solution:

= 5x × (5x2 + 2x + 6)

=  (5x × 5x2) + (5x × 2x) + (5x × 6)

= 25x3 + 10x2 + 30x

Polynomial Multiplied by a Polynomial

To multiply a polynomial and a monomial we need to multiply each and every term of one polynomial with each and every term of other polynomials.

Examples: Find the product of (5x2 + 2x + 6) and (x2 + 2x + 3) 

Solution:

= (5x2 + 2x + 6) × (1x2 + 2x + 3) 

= (5x2 × 1x2) + (5x2 × 2x) + (5x2 × 3) + (2x × 1x2) + (2x × 2x) + (2x × 3) + (6 × 1x2) + (6 × 2x) + (6 × 3)

= 5x4 +10x3 + 15x2 + 2x3 + 4x2 + 6x + 6x2 + 12x + 18

= 5x4 +12x3 + 21x2 + 18x + 18

Read More,

Multiplying Polynomials Examples

Example 1: Find the product of (3x2 + 1x + 2) and (1x2 + 2x + 1)

Solution:

= (3x2 + 1x + 2) × (1x2 + 2x + 1) 

= (3x2 × 1x2) + (3x2 × 2x) + (3x2 × 1) + (1x × 1x2) + (1x × 2x) + (1x × 1) + (2 × 1x2) + (2 × 2x) + (2 * 1)

= 3x4 +6x3 + 3x2 + 1x3 + 2x2 + 1x + 2x2 + 4x + 2

= 3x4 +7x3 + 7x2 + 5x + 2

Example 2: Find the product of (5xy + 1) and (2z + 3)

Solution:

= (5xy + 1) × (2z + 3)

= (5xy × 2z) + (5xy × 3) + (1 × 2z) + (1 × 3)

= 10xyz + 15xy + 2z + 3

Example 3: Find the Product of (3xyz) and (2x + 6)

Solution:

= (3xyz) × (2x + 6)

= (3xyz × 2x) + (3xyz × 6)

= 6x2yz +18xyz

Example 4: Find the product of (−a3b) and (2ab3)

Solution:

= (−a3b) × (2ab3)

= -2a4b4

Example 5: Find the product of (xy + 2y) and (a + b)

Solution:

= (xy + 2y) × (a + b)

= (xy × a) + (xy × b) + (2y × a) + (2y × b)

= axy + bxy + 2ay + 2by

Practice Questions on Multiplying Ploynomial

Q1. Multiply 2x2 and 3xy

Q2. Multiply (3x2 – 5y) and (4x – y)

Q3. Multiply (x + 2y) and (3x2 − 4xy + 5)

Q4. Multiply (xy – 3) and (2x2 – 9y)

FAQs on Multiplying Polynomials

1. What is a Polynomial?

A polynomial is a algebraic expression formed by multiplying various variables and constant together.

2. How to Multiply Polynomials?

To multiply polynomials we multiply first term of the first polynomial with each term of the second polynomial and then repeat the process for all the terms.

3. What are 3 ways to Multiply Polynomials?

Multiplication of Polynomial is achieved using the following 3 methods,

  • FOIL Method
  • Box Method
  • Distributive Property Method\

4. What is FOIL method of Multiplying Polynomial?

A foli method is a method of multiplication of polynomial, it is an abbreviation for First Outer Inner the ast



Last Updated : 06 Sep, 2023
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