Polynomial Expressions - Definition, Degree, Examples

Polynomial expressions are one of those algebra topics that almost every student meets early on and then keeps using again and again. At first, they may look confusing, but once you understand the basic rules, they actually become quite easy to work with. A polynomial expression is made using variables, numbers, and exponents, all connected using addition, subtraction, or multiplication.

Polynomial Expressions 

When algebra starts, polynomial expressions show up everywhere. A polynomial is simply made up of terms. Each term can be a number, a variable, or both together. For example, in the expression 3x2+2x−53x^2 + 2x - 53x2+2x−5 each part separated by a plus or minus sign is a term.

Polynomial expressions help turn real-world problems into math problems. Things like speed, height, area, or profit can all be written using polynomials. This is why teachers spend so much time on this topic.

The word polynomial comes from two parts: “poly,” which means many, and “nomial,” which means terms. In the example above, x is the variable, 3 and 2 are coefficients, and −5 is a constant.

Keep in mind that expressions like 1/x1/x1/x or x−1x^{-1}x−1 are not polynomial expressions. The moment a variable goes into the denominator or gets a negative power, it stops being a polynomial.

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Definition of Polynomial Expressions

To decide whether an expression is a polynomial or not, you just need to check a few basic rules. This polynomial expressions definition requires that every exponent of a variable must be a whole number like 0, 1, 2, or 3. If you see a fractional or negative exponent, it is not a polynomial. Variables should also not appear under a root sign or in the denominator. All numbers used must be real numbers.

Polynomial expressions are often grouped based on how many terms they have:

• Monomial – one term
  
  

• Binomial – two terms
  
  

• Trinomial – three terms
  
  

This way, solving problems will be easier and it will help the pupils to have an organized work.

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Polynomial Expressions Examples

You come across polynomial expressions almost on a daily basis whenever you explore mathematics. Here are a few:

• 5x+35x + 35x+3, which is a linear polynomial
  
  

• 4x2−94x^2 - 94x2−9, which is a quadratic polynomial
  
  

Here are a few more examples:

• Monomial: 7xy7xy7xy
  
  

• Binomial: x2−4x^2 - 4x2−4
  
  

• Trinomial: ax2+bx+cax^2 + bx + cax2+bx+c
  
  

• Constant polynomial: 7
  
  

It is termed as the term containing the leading term of the variable. Polynomials can be written in standard form, where each term is, in an organized manner, from highest to lowest power of the variable, making them simpler to comprehend and solve.

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Degree of Polynomial Expressions

The degree of a polynomial simply means the highest power of the variable in the expression. For example, in

x3+x2+1x^3 + x^2 + 1x3+x2+1

the degree is 3.

The degree tells us a lot about a polynomial. It helps us understand how graphs behave and how many solutions an equation can have. A constant polynomial, like 6 or −10, has a degree of zero.

There is one special case you should remember. The zero polynomial, which is just 0, does not have a degree. Its degree is said to be undefined, and this often appears as a tricky question in tests.

If a term has more than one variable, such as 3x2y33x^2y^33x2y3, you add the powers to find the degree. So here, the degree is 2+3=52 + 3 = 52+3=5.

Polynomial Expressions Worksheets

The best way to get comfortable with polynomial expressions is by practicing regularly. Worksheets help you spot patterns and quickly identify which expressions are polynomials and which are not. Whenever you see fractional powers or variables in denominators, you can immediately say that the expression is not a polynomial.

Another important idea is the leading coefficient, which is the number in front of the highest-degree term. In

8x4+2x8x^4 + 2x8x4+2x

the leading coefficient is 8.

The degree of a polynomial also gives a rough idea of how many solutions an equation might have. This becomes useful later when solving polynomial equations.

Easy Rules to Remember

• Exponents must be whole and non-negative
  
  

• Variables cannot be under root signs
  
  

• Variables cannot be in denominators
  
  

• The degree is the highest power present
  
  

• Standard form goes from highest power to lowest

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How to Find the Degree with Two Variables

When a term has two variables, finding the degree is simple. Just add the powers. For example:

4x2y24x^2y^24x2y2

The degree is 2+2=42 + 2 = 42+2=4.

If the full polynomial is

4x2y2+3x4x^2y^2 + 3x4x2y2+3x

The degree of the polynomial is 4, since that is the highest degree among all terms.

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