Polynomial multiplication is the process of multiplying two or more polynomials to find their product. It involves multiplying coefficients and applying exponent rules for variables.
When multiplying polynomials:
- Multiply the coefficients (numerical values).
- Multiply variables with the same base by adding their exponents.
- Write different variables together as they are.
- Combine like terms if needed.
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.
Multiplying 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.
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.
Degree of Polynomials
The degree of polynomial after multiplying two polynomials will be always greater than the degree of the individual polynomials.
Degree (A × B) = Degree(A) + Degree(B)
Steps to Multiply Polynomials
The steps related to Multiplying polynomials are :
Step 1: Multiply each term of one polynomial with each term in the other polynomial by using the distributive law.
Step 2: Sum up the powers of the same variables using the law of exponents.
Step 3: Combine the like term by adding or subtracting.
Methods to Multiply 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 monomial is different, etc.
There can be many possible cases involving monomial, binomial, or polynomials while multiplying. Some of the common cases are:
- Multiplying Polynomials with Exponents
- Multiplying Polynomials with Different Variables
- Multiplying a Monomial by a Polynomial
- Multiplying a Polynomial by a Polynomial
- Multiplying a Monomial by a Monomial
- Multiplying a Binomial by a Binomial
- Multiplying Binomials by Box Method
Let's discuss these cases in detail as follows:
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 coefficient 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 a Monomial 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
Multiplying a Polynomial by a Polynomial
To multiply two polynomials, 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
Multiplying a Monomial 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
Multiplying a Binomial by a Binomial
Two binomials can be multiplied using the distributive properties. The distributive properties of the algebra are,
(a + b) · (c + d) = a · c + a · d + b · c + b · d
Using this property we can easily multiply two binomial. 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
Multiplying Binomials by Box Method
The box method (or area model) is a visual way to multiply polynomials, particularly binomials. It helps organize calculations and makes it easy to combine like terms.
Steps to Multiply Binomials using the Box Method
- Set up a 2 × 2 grid (box):
- Write the terms of the first binomial along the top.
- Write the terms of the second binomial along the left side.
- Multiply each row and column entry:
- Multiply each term from the row with each term from the column.
- Fill in the grid with the products.
- Combine like terms (if any).
Example: Multiply (2x2 + 2) (x + 2x)
\begin{array}{|c|c|c|}\hline \times & 2x^{2} & + 2\\\hline x & 2x^{3} & 2x \\ \hline +2x & 4x^{3} & 4x \\ \hline \end{array}
Combine like terms:
2x3 + 4x3 + 2x + 4x
6x3 + 6x
Thus, (2x2 + 2) (x + 2x) = 6x3 + 6x
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Solved Examples of Multiplying Polynomials
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 Polynomial
Question 1: Multiply 2x2 and 3xy.
Question 2: Multiply (3x2 – 5y) and (4x – y).
Question 3: Multiply (x + 2y) and (3x2 − 4xy + 5).
Question 4: Multiply (xy – 3) and (2x2 – 9y).
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