Polynomial Expressions

The word polynomial is derived from two parts: ‘poly,’ a Greek word meaning ‘many,’ and ‘nomial,’ meaning ‘term.’ Together, a polynomial means many terms.

In simple words, A polynomial is an expression that involves addition, subtraction, and multiplication but not division. Each term includes a variable raised to a whole-number exponent and multiplied by a coefficient.

A polynomial can be classified into several types based on the number of terms it contains. They are:

• Monomial 

• Binomial

• Trinomial

The following table provides the types of polynomials on the basis of number of terms.

The degree of the polynomial is the highest power of the variable in the expression. 

Polynomial expressions are classified into several types based on their degree. 

Simplifying a polynomial expression means combining like terms and rewriting it in a simpler form to make calculations easier. Let understand this using step-by-step breakdown of simplifying a polynomial.

For example, 4x² + 2x + 7 + 3x + 2x² - x - 4

1. Group-like terms:
   (4x² + 2x²) + (2x + 3x - x) + (7 - 4)
    
2. Add or subtract the coefficients:
   (4 + 6)x² + (2 + 2 - 1)x + (3)
    
3. Simplified Expression:
   6x² + 4x + 3

Let's practice this using the given problem.

Practice Problem: 2x² - 3x³ + 5x³ - 4 - 7x + 20

Solution:

1. Group-like terms:
   (-3x³ + 5x³) + (2x²) - 7x  + (- 4 + 20)
    
2. Add or subtract the coefficients:
   (-3 + 5)x³ + 2x² - 7x + 16
    
3. Simplified Expression:
   2x³ + 2x² - 7x+ 16

Parent Tip: You can use real life examples to explain like and unlike terms. Such as 2 apples and 3 tangerines are unlike terms and cannot be added. But 2 roses and 4 roses are like terms and can be added.

For better understanding and to make calculations easy, here are as few tips and tricks that will help you master polynomial expression.

1. Add or subtract only like terms. For example, 3x and 5 cannot be added, but 3x and 1x can.
    
2. Remember monomial, binomial, and trinomial polynomials from its words. Mono means ones, implies monomial has 1 term, bi means pair, implies binomial has two terms, and tri means three, that gives trinomial has three terms.
    
3. Always write the final answer in simplified form.
    
4. Carefully perform the arithmetic operations.
    
5. To identify the type of polynomial, check its degree.

Parent Tip: Use combining like terms' calculator to check your child's calculation. Encourage your child to practice by solving different problems.

Polynomials are not limited to classroom studies; they are also used in our daily lives, often without us even realizing it. Here are some real-life applications of polynomial expressions:

1. Designing Roller Coasters: Engineers use polynomials to create smooth curves, so the ride is safe and fun and also safe. The path of the ride is also expresses using polynomials.
   
    
2. Moving Vehicles: Mathematicians and car designers use polynomial expressions to predict how far a car will travel based on how long and how fast it moves.
   
    
3. Science and Mathematics: Polynomials are used in data analysis to analyze trends and make predictions in various fields.
   
    
4. Computer Graphics: Polynomial expressions are used to create, manipulate, and animate curves, surfaces, and transformations. The jumping of a character or throwing of a ball in any video game is also modelled using polynomial expressions.
   
    
5. Robotics: In robotics, polynomials are used for programming and controlling robot's movements and rotations.