To multiply polynomials, first, multiply each term in one polynomial by each term in the other polynomial using distributive law. Then, simplify the resulting polynomial by adding or subtracting the like terms. It should be noted that the resulting degree after multiplying two polynomials will be always more than the degree of the individual polynomials.
How to Multiply Polynomials?
Follow the below-given steps for multiplying polynomials:
- Step 1: Place the two polynomials in a line.
For example, for two polynomials, (6xâˆ’3y) and (2x+5y), write as: (6xâˆ’3y)Ã—(2x+5y)
- Step 2: Use distributive law and separate the first polynomial.
â‡’ (6xâˆ’3y)Ã—(2x+5y) = [6x Ã— (2x+5y)] âˆ’ [3y Ã— (2x+5y)]
- Step 3: Multiply the monomials from the first polynomial with each term of the second polynomial.
â‡’ [6x Ã— (2x+5y)] âˆ’ [3y Ã— (2x+5y)] = (12x^{2}+30xy) âˆ’ (6yx+15y^{2})
- Step 4: Simplify the resultant polynomial, if possible.
â‡’ (12x^{2}+30xy) âˆ’ (6yx+15y^{2}) = 12x^{2}+24xyâˆ’15y^{2}
Points to Note:
When multiplying polynomials, the following pointers should be kept in mind:
- Distributive Law of multiplication is used twice when 2 polynomials are multiplied.
- Look for the like terms and combine them. This may reduce the expected number of terms in the product.
- Preferably, write the terms in the decreasing order of their exponent.
- Be very careful with the signs when you open the brackets.
Resultant Degree after Multiplying Polynomials
For two polynomials equations, P and Q, the degree after multiplication will always be higher than the degree of P or Q. The degree of the resulting polynomial will be the summation of the degree of P and Q.
So,
Degree (P Ã— Q) = Degree(P) + Degree(Q) |
Topics Related to Polynomial Multiplication
Types of Polynomial Multiplication:
It is known that there are different types of polynomial based on its degree like linear, binomial, quadratic, trinomial, etc. The steps to multiply polynomials is same for all the types. Here, two types of multiplication of polynomials are explained in detail.
- Multiplication of Binomial by a Binomial
- Multiplication of a Binomial by a Trinomial
Multiplying Binomial by a Binomial
When a binomial is multiplied with a binomial, the distributive law of multiplication is followed.
We know that Binomial have 2 terms. Multiplying two binomials give the result having a maximum of 4 terms (only in case when we donâ€™t have like terms). In case of like terms, the total number of terms is reduced.
Like Terms:
According to the commutative law of multiplication, terms like ab and ba gives the same result. Thus they can be written in both the forms.
For example, 5Ã—6 = 6Ã—5 = 30
Now, Consider two binomials given as (a+b) and (m+n).
Multiplying them we have,
(a+b)Ã—(m+n)
â‡’ aÃ—(m+n)+bÃ—(m+n) (Distributive law of multiplication)
â‡’ (am+an)+(bm+bn) (Distributive law of multiplication)
Thus,
(a + b) Ã— (m + n) = am + an + bm + bn |
Example 1: Find the result of multiplication of two polynomials (6x +3y) and (2x+ 5y).
Solution- (6xâˆ’3y)Ã—(2x+5y)
â‡’6xÃ—(2x+5y)âˆ’3yÃ—(2x+5y) (Distributive law of multiplication)
â‡’(12x^{2}+30xy)âˆ’(6yx+15y^{2}) (Distributive law of multiplication)
â‡’12x^{2}+30xyâˆ’6xyâˆ’15y^{2} (as xy = yx)
Thus, (6x+3y)Ã—(2x+5y)=12x^{2}+24xyâˆ’15y^{2}
Example 2:
Let us take up an example. Say, you are required to multiply a binomial (5y + 3z) with another binomial (7y âˆ’ 15z). Let us see how it is done.
(5y + 3z) Ã— (7y âˆ’ 15z)
= 5y Ã— (7y âˆ’ 15z) + 3z Ã— (7y âˆ’ 15z) (Distributive law of multiplication)
= (5y Ã— 7y) âˆ’ (5y Ã— 15z) + (3z Ã— 7y) âˆ’ (3z Ã— 15z) (Distributive law of multiplication)
= 35y^{2} âˆ’ 75yz + 21zy âˆ’ 45z^{2}
= 35y^{2} âˆ’ 75yz + 21yz âˆ’ 45z^{2}
As, (yz = zy)
(5y + 3z) Ã— (7y âˆ’ 15z) = 35y^{2} âˆ’54yz âˆ’ 45z^{2}
Multiplying Binomial with a Trinomial
When multiplying polynomials, that is, a binomial by a trinomial, we follow the distributive law of multiplication. Thus, 2 Ã— 3 = 6 terms are expected to be in the product. Let us take up an example.
(a^{2} âˆ’ 2a) Ã— (a + 2b âˆ’ 3c)
= a^{2} Ã— (a + 2b âˆ’ 3c) âˆ’ 2a Ã— (a + 2b âˆ’ 3c) (Distributive law of multiplication)
= (a^{2} Ã— a) + (a^{2} Ã— 2b) + (a^{2} Ã— âˆ’3c) âˆ’ (2a Ã— a) âˆ’ (2a Ã— 2b) âˆ’ (2a Ã— âˆ’3c) (Distributive law of multiplication)
= a^{3} + 2a^{2}b âˆ’ 3a^{2}c âˆ’ 2a^{2} âˆ’ 4ab + 6ac
Now, by rearranging the terms,
(a^{2} âˆ’ 2a) Ã— (a + 2b âˆ’ 3c) = a^{3} âˆ’ 2a^{2} + 2a^{2}b âˆ’ 3a^{2}câˆ’ 4ab + 6ac