Multiplication Of A Polynomial By A Polynomial: Definition, Rules

While solving algebraic equations you must have come across sums which involve multiplication of a polynomial by a polynomial. It is one of the fundamental concepts of algebra. Initially it might seem to be tougher than it really is. There are certain methods and steps required to make the calculations easier. In this article you will find a detailed explanation of the methods and the steps involved in the multiplication of a polynomial by a polynomial.

Key terms:- polynomial, binomial , monomial, exponential law, distributive law

Also read: Isosceles Triangle Theorems

---

Rules for Multiplying Polynomials

[Click Here for Sample Questions]

1. Multiply each term of one polynomial to each term of the other polynomial using the distributive law.
2. Add the powers of the same variable using the exponential law.
3. Then, simplify the obtained equation by adding or subtracting the like terms.

It should be kept in mind that the resulting degree of the polynomial will always be more than the degree of the given individual polynomials. If the degree of one or more multiplying polynomials is in minus powers then only the degree of the resulting polynomial is less than the degree of the obtained polynomial.

The video below explains this:

Polynomials Detailed Video Explanation:

---

Multiplying a Polynomial By a Polynomial Using Exponential Law

[Click Here for Sample Questions]

Though any multiplication of a polynomial by another polynomial involves the use of distributive law of multiplication (except for multiplication involving only monomials). In some special cases, where the polynomials contain like variables, the exponential law is applied to make the calculations easier.

For example:-

• (25x ³)×(x ⁴)= (25)(x ³⁺⁴)= 25x ⁷
• (7x² y²)× (4x³ y ³)= (7×4)(x ²⁺³)(y²⁺³)= (28x ⁵y ⁵)
• (15x⁶) (4x^-4) = (15×4) (x^6-4) = 60x²

Multiplying Polynomials By Polynomials

Usually, there are mainly three types of polynomials , namely monomials, binomials and trinomials.

The steps involved in the multiplication of the above mentioned polynomials are the same.

Multiplication of a Monomial by a Monomial

A monomial is a single term polynomial. Multiplication of a monomial by a monomial involves simple multiplication of the coefficients and the multiplication of variables together using exponential law (if the variables are the same).

For example:-

• 2x × 4y = (2×4)(x × y) = 8xy
• 3x × 5y × 6z = (3×5×6)(x × y × z)
• 2x² × 5x³ = (2×5)(x ² × x 3) = (10)(x ²+3) = 10x ⁵

Multiplication of a Binomial by a Binomial

A polynomial having two terms is called a binomial. Multiplication of a binomial by a binomial involves the use of distributive law  and incase of like variables, the exponential law as well.

When the two given binomials do not have the like terms, then there are a maximum of 4 terms in the resulting answer. When there are like terms in the given binomials then the number of terms of the result is less than 4.

For example:-

• Two like terms-

(2x + 4y) (3x + 3y)

Using the distributive law-

= 2x(3x + 3y)+ 4y(3x + 3y)…..

 = 6x² + 6xy + 12xy + 12y²

Adding the like terms-

=6x² + 18xy + 12y²

• Four like terms-

(6a – b) (3x - 4y)

= 6a (3x - 4y) - b (3x - 4y)

=18ax – 24ay - 3bx + 4by

Multiplication of a Binomial by a Trinomial

A polynomial having three terms is known as a trinomial. Multiplication of a binomial by a trinomial involves the same steps but one should be very careful with the change of signs.

The maximum number of terms obtained in the answer is 6, but if the polynomials contain like terms then it can be lesser.

Also read :- PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

For example,

(a²- 3b²)(4a + 5a² - b)

Using the distributive law-

=a²(4a + 5a² - b) - 3b²(4a + 5a² - b)

=4a³ + 5a³ - a²b - 12ab² - 15a²b² + 3b³

Adding the like terms-

= 9a³ - a²b - 12ab²- 15a²b² + 3b³

Read More:

---

Things to Remember

• When two polynomials are multiplied the distributive law of multiplication is used twice.
• When the given polynomials have unlike terms only then the multiplication is done by using only the Distributive Law of Multiplication
• When there are like terms in the polynomials, their powers can be added using the exponential law.
• Like terms must be added or subtracted accordingly to reduce the total number of terms in the result.
• Try to write the terms in the decreasing order of their exponents.
• Be very careful with the signs while opening the brackets.

Also Read: