Multiplying Polynomials

Polynomials are algebraic expressions consisting of constants and variables that are combined using mathematical operations like addition, subtraction, and multiplication. This method is based on the number of terms and their degree.

Listed below are some important rules to understand while multiplying polynomials:

Multiply the coefficients

When multiplying variables with the same base, just add their exponents.

For different variables, simply list them next to each other; they cannot be added together.

Combine all like terms after multiplying to simplify the expression.

The steps followed for the multiplication of polynomials are explained below using examples.

Question: Multiply: (x+2)(x+3)

Solution: 

Step 1: Distribute each term:
 x(x+3)+2(x+3)
Step 2: Multiply each part:
x2+3x+2x+6
Step 3: Combine like terms 
x2+5x+6

Answer: (x+2)(x+3)=x2+5x +6
So, the steps involved in the multiplication of polynomials are:
Multiply each term in the first polynomial with every term in the second polynomial.
Multiply the coefficients and variables 
Combine like terms

Multiplying polynomials can differ depending on the form of the polynomials, like monomials, binomials, or polynomials having higher degrees.

Given below are several scenarios requiring different methods of multiplication. 

• 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 the Box Method

This method is used to multiply polynomials having the same variables but different exponents. In this case, we follow the given steps:
 

Step 1: Multiply the coefficients of both variables.
 

Step 2: Use the law of exponents to multiply the variables.

For example: Multiply (3x²)(4x⁵)

Solution: multiply the coefficients 3 × 4 = 12
Use the law of exponents for the same base: x² × x⁵ = x²⁺⁵ = x⁷

So, (3x²)(4x⁵) =12x⁷

While multiplying polynomials that have different variables like x, y, or a, b we follow the given steps:

Step 1: Multiply the coefficients

Step 2: Write different variables side by side, do not add the exponents since the variables are not the same.

Step 3: Apply the distributive property for more than one term in the polynomial.

Step 4: Combine like terms.
For example: Multiply (2x+3y)(4x+y)
Solution: 

Use the distributive property to multiply each term in the first polynomial with each term of the second polynomial:

 2x . 4x+2x . y+3y . 4x+3y . y

Multiply coefficients and variables, and write the different variables side by side: 
2x . 4x=8x2
2x . y=2xy
3y . 4x=12xy
3y . y=3y2
8x²+2xy+12xy+3y²

Combine like terms:
2xy+12xy=14xy

Answer: 8x²+14xy+3y²

For multiplying 2 monomials:

Step 1: Multiply both variables by each other.

Step 2: Multiply the coefficients of both monomials. 

Let's take two monomials: 4a² and 5a³

Multiplying both variables, we get: a² × a³ = a²⁺³ = a⁵

Multiplying both coefficients gives us: 4 × 5 = 20

So, 4a² × 5a³ = 20a⁵

To multiply three monomials together  (2x²)(−3x⁴)(5y):

Step 1: Multiply the coefficients of all the terms.
2 × (-3) × 5 = -30

Step 2: Multiply all the variables together
x² × x⁴ × y = x²⁺⁴y = x⁶y

So, (2x²)(−3x⁴)(5y) = - 30x⁶y
 

Binomials are polynomials containing two terms. Binomials can be multiplied by other binomials using two methods:
The distributive property or FOIL method, and
Box method
The distributive method: Multiply the first binomial by the second by distributing one term at a time and combining the results.
For example: (2x + 3) (4x + 5)
Apply the distributive property:
(2x + 3)(4x+5)=2x(4x+5)+3(4x+5)
Distribute each term
2x . 4x+2x . 5+3 . 4x+3 . 5
= 8x2+10x+12x+15
Combine like terms:
=8x²+ (10x+12x)+15
= 8x²+22x+15
So,  (2x + 3) (4x + 5) = 8x²+22x+15

The box method: A 2 × 2 grid is created with one binomial written across the top and the other down the side. The empty boxes left are filled with the products of terms in the row and column of that intersecting box. Then all the terms are added.
For example: (x + 7)(x + 3)

      x   7
+   x2  7x
 x 
  3  3x  21

Now add all the terms: x² + 7x + 3x + 21 = x² + 10x + 21
 

We use the distributive property to multiply a monomial by a binomial. Using an example lets us understand how to apply the distributive property in this case:
 

Question: Multiply (4a)(2b + 7c)

Solution: 4a × 2b + 4a × 7c
= (4 × 2 ×  a ×  b) + (4 × 7 × a  × c)
= 8ab + 28ac

Polynomials are used to model quantities that change with time or input. These quantities include area, speed, and profit. Some real-life applications of polynomial multiplication are listed below:
 

• To expand and simplify expressions

When we multiply polynomials, we turn complex expressions like (x + 2) (x + 3) into a simpler, single expression:
X2 + 5x + 6. This makes it easier to solve, graph, or analyze.

• To model real-life problems in architecture, economics, and agriculture

Multiplying polynomials helps model real situations like profits, growth, motion, or area, where more than one variable is changing. For example, a farmer’s yield may depend on fertilizer and water. Each affects the output differently. We can see the total impact by multiplying polynomials.

• To build new functions

Sometimes we need new formulas from old ones. Multiplying polynomials gives us a bigger, more complete formula for making predictions.

• To Prepare for Solving Equations

In algebra and other fields, many problems require multiplying polynomials before solving for unknowns.

• Multiplying coordinate expressions for curves

In animation, 3D modeling, and game design, polynomial multiplication helps get new shapes or curves for scaling, rotating, and transforming objects.