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.

\(3x^3 -4x^2 + 7x + 18\) is an example of polynomial.

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

1. Multiply the coefficients
    
2. When multiplying variables with the same base, just add their exponents.
    
3. For different variables, simply list them next to each other; they cannot be added together.
    
4. 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:
  \(x^2+3x+2x+6\)
   
• Step 3: Combine like terms 
  \(x^2+5x+6\)

Answer: \((x+2)(x+3)=x^2+5x +6\)

So, the steps involved in the multiplication of polynomials are:

1. Multiply each term in the first polynomial with every term in the second polynomial.
    
2. Multiply the coefficients and variables 
    
3. Combine like terms

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

• Monomial: A polynomial having one term is called a monomial.
• Binomial: A polynomial that only consists of two terms is called a binomial.

Given below are several scenarios requiring different methods of multiplication. 

1. Multiplying Polynomials with Exponents
2. Multiplying Polynomials with Different Variables
3. Multiplying a Monomial by a Polynomial
4. Multiplying a Polynomial by a Polynomial
5. Multiplying a Monomial by a Monomial
6. Multiplying a Binomial by a Binomial
7. 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.

Study Strategy: The laws of exponents can be studied from the following table

 

Let's practice this using an example.
 

For example: Multiply (3x²)(4x⁵)

Solution: 

• Multiply the coefficients 3 × 4 = 12
• Use the law of exponents for the same base: \(x^2 \times x^5 = x^{2+5} = x^7\)
   

So, \((3x^2)(4x^5) =12x^7\)

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.
  
  The distributive property says that:
  
  
   
• Step 4: Combine like terms.

For example: Multiply (2x + 3y)(4x + y)
Solution: 

1. Use the distributive property to multiply each term in the first polynomial with each term of the second polynomial:
   
   \( 2x . 4x+2x . y+3y . 4x+3y . y\)
    
2. Multiply coefficients and variables, and write the different variables side by side: 
   
   \(2x . 4x=8x^2\\ 2x . y=2xy\\ 3y . 4x=12xy\\ 3y . y=3y^2\\ 8x^2+2xy+12xy+3y^2\)
    
3. Combine like terms:
   
   \(2xy+12xy=14xy\)

Answer: \(8x^2+14xy+3y^2\)

Monomial consists of only one term. Examples are 2, 3x³, 8a², 3, etc

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

1. Multiplying both variables, we get: a² × a³ = a²⁺³ = a⁵
2. Multiplying both coefficients gives us: 4 × 5 = 20
   So, 4a² × 5a³ = 20a⁵

Multiplying three monomials

To multiply three monomials (2x²)(−3x⁴)(5y) together, follow the steps: 

• Step 1: Multiply the coefficients of all the terms.
  2 × (-3) × 5 = -30
   
• Step 2: Multiply all the variables together
  \(x^2 × x^4 × y = x^2+4y = x^6y\)

So, \((2x^2)(−3x^4)(5y) = - 30x^6y\)
 

The total number of terms in a binomial are only two. Binomials can be multiplied by other binomials using two methods:

1. The distributive property or FOIL method, and
2. 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)
   

1. Apply the distributive property:
   \((2x + 3)(4x+5)=2x(4x+5)+3(4x+5)\)
    
2. Distribute each term
   \(2x . 4x+2x . 5+3 . 4x+3 . 5\\ = 8x^2+10x+12x+15\)
    
3. Combine like terms:
   \(=8x^2+ (10x+12x)+15\\ = 8x^2+22x+15\)
   
   So,  \((2x + 3) (4x + 5) = 8x^2+22x+15\)

• 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)
  
  
  
  Now add all the terms: \(x^2 + 7x + 3x + 21 = x^2 + 10x + 21\)

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

To practice multiplication, let's take a quiz.

Quiz: Multiply (2x)(3y + 8)

Answer: 6xy + 16x

Students from smaller grades can find multiplying polynomials difficult in the beginning. So, let's focus some tips and tricks to make multiplication of polynomials easy:

1. You can use online calculators for combining like terms and to verify your answer.
    
2. Learn basic exponents rules.
    
3. Remember, only like terms can be added and subtracted.
    
4. Properly apply the distributive property. Each term of a polynomial should be multiplied to every term of another polynomial.
    
5. Perform the multiplication carefully. Practice problems from multiplying polynomials worksheet.

Parent Tip: Encourage your child to practice problems. You can verify their answer through multiplying polynomials calculator.
 

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:
 

1. 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.
   
    
2. Physics
   Quantities such as distance, speed, and energy varies with respect to time. To find parameters like kinetic energy, you need to multiply polynomials.
   
    
3. Probability
   Multiplication of polynomials is used to compute probability for various outcomes, for example in binomial theorem.
   
    
4. Computer Graphics
   In animation, 3D modeling, and game design, polynomial multiplication helps get new shapes or curves for scaling, rotating, and transforming objects.
   
    
5. Biology
   The growth of bacteria is modelled using polynomial functions for scientific studies. To calculate the rate of growth, the polynomial functions are multiplied.