Multiplying Polynomials

Multiplying polynomials is a basic concept in algebra. Multiplication of two polynomials will include the product of coefficients to coefficients and variables to variables. We can easily multiply polynomials using rules and following some simple steps. Let us learn more about multiplying polynomials with examples in this article.

Rules for Multiplying Polynomials

Multiplying polynomials require only three steps.

• First, multiply each term in one polynomial by each term in the other polynomial using the distributive law.
• Add the powers of the same variables using the exponent rule.
• 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.

Multiplying Polynomials Using Exponent Law

If the variable is the same but has different exponents of the given polynomials, then we need to use the exponent law.

Example: Multiply 2x² × 3x

Here, the coefficients and variables are multiplied separately.

= (2 × 3) × (x² × x)

= 6 × x²⁺¹

= 6x³

Multiplying Polynomials having different variables

Follow the below-given steps for multiplying polynomials:

• Step 1: Place the two polynomials in a line.

For example, for two polynomials, (6x−3y) and (2x+5y), write as: (6x−3y)×(2x+5y)

• Step 2: Use distributive law and separate the first polynomial.

⇒ (6x−3y)×(2x+5y) = [6x × (2x+5y)] − [3y × (2x+5y)]

• Step 3: Multiply the monomials from the first polynomial with each term of the second polynomial.

⇒ [6x × (2x+5y)] − [3y × (2x+5y)] = (12x²+30xy) − (6yx+15y²)

• Step 4: Simplify the resultant polynomial, if possible.

⇒ (12x²+30xy) − (6yx+15y²) = 12x²+24xy−15y²

Degree – 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,

Multiplying Polynomials by Polynomials

It is known that there are different types of polynomial based on their degree like monomial, binomial, trinomial, etc. The steps to multiply polynomials are the same for all types.

Multiplying Monomial by Monomial

A monomial is a single term polynomial. If two or more monomials are multiplied together, then the resulting product will be a monomial.

Examples are:

• 5x × 7x = 5 × x × 7 × x = 35x²
• 2x × 3y × 4z = 2 × x × 3 × y × 4 × z = (2 × 3 × 4) × (x × y × z) = 24xyz

Multiplying Binomial by a Binomial

A binomial is a two-term polynomial. When a binomial is multiplied by a binomial, the distributive law of multiplication is followed.

We know that Binomial has 2 terms. Multiplying two binomials give the result having a maximum of 4 terms (only in case when we don’t have like terms). In the 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’ give the same result. Thus they can be written in both forms.

For example, 5×6 = 6×5 = 30

Now, Consider two binomials given as (a+b) and (m+n).

Multiplying them we have,

(a+b)×(m+n)

⇒ a×(m+n)+b×(m+n) (Distributive law of multiplication)

⇒ (am+an)+(bm+bn) (Distributive law of multiplication)

Thus,

Solved Examples

Example 1: Find the result of multiplication of two polynomials (6x +3y) and (2x+ 5y).

Solution- (6x−3y)×(2x+5y)

⇒6x×(2x+5y)−3y×(2x+5y) (Distributive law of multiplication)

⇒(12x²+30xy)−(6yx+15y²) (Distributive law of multiplication)

⇒12x²+30xy−6xy−15y² (as xy = yx)

Thus, (6x+3y)×(2x+5y)=12x²+24xy−15y²

Example 2:

Let us take up an example. Say, you are required to multiply a binomial (5y + 3z) with another binomial (7y − 15z). Let us see how it is done.

(5y + 3z) × (7y − 15z)
= 5y × (7y − 15z) + 3z × (7y − 15z) (Distributive law of multiplication)

= (5y × 7y) − (5y × 15z) + (3z × 7y) − (3z × 15z) (Distributive law of multiplication)

= 35y² − 75yz + 21zy − 45z²

= 35y² − 75yz + 21yz − 45z²

As, (yz = zy)

(5y + 3z) × (7y − 15z) = 35y² −54yz − 45z²

Multiplying Binomial with a Trinomial

A trinomial is a three-term polynomial. When multiplying polynomials, that is, a binomial by a trinomial, we follow the distributive law of multiplication. Thus, 2 × 3 = 6 terms are expected to be in the product. Let us take up an example.

(a² − 2a) × (a + 2b − 3c)

= a² × (a + 2b − 3c) − 2a × (a + 2b − 3c) (Distributive law of multiplication)

= (a² × a) + (a² × 2b) + (a² × −3c) − (2a × a) − (2a × 2b) − (2a × −3c) (Distributive law of multiplication)

= a³ + 2a²b − 3a²c − 2a² − 4ab + 6ac

Now, by rearranging the terms,

(a² − 2a) × (a + 2b − 3c) = a³ − 2a² + 2a²b − 3a²c− 4ab + 6ac

Important Facts

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.

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Practice Questions

• Multiply 2x by 3y.
• Multiply (3x – a) (4x – y)
• Find the product of (x + 2y)(3x − 4y + 5)
• Multiply (x – 3) (2x – 9)
• What is the product of 3x² and 4x²– 5x + 7?