Polynomials are types of expressions. You can think of polynomials as a dialect of mathematics. They are used to express numbers in almost every field of mathematics and are considered very important in certain branches of math, such as Calculus. For example, 2x + 9 and x2 + 3x + 11 are polynomials. You might have noticed that none of these examples contain the "=" sign. Have a look at this article in order to understand polynomials in a better way.
|1.||What is a Polynomial?|
|2.||Like Terms and Unlike Terms|
|3.||Degree of a Polynomial|
|4.||Standard Form of Polynomials|
|5.||3 Types of Polynomials|
|6.||Addition and Subtraction of Polynomials|
|7.||FAQs on Polynomials|
What is a Polynomial?
A polynomial is a type of expression. Before we discuss polynomials, we should know about expressions. What is an expression? An expression is a mathematical statement without an equal-to sign (=). Coming back to polynomials, the definition of the polynomial can be given as: "A polynomial is a type of expression in which the exponents of all variables should be a whole number." Let us understand this by taking an example: 3x2 + 5. In the above polynomial, there are certain terms that we need to understand. Here, x is known as the variable. 3 which is multiplied to x2 has a special name. We denote it by the term "coefficient." 5 is known as the constant. The power of the variable x is two.
Note: The powers of the variables in any polynomial has to be a non-negative integer.
The following image shows all the terms in a polynomial.
Like Terms and Unlike Terms
Like terms in polynomials are those terms which have the same variable and same power. Terms that have different variables and different powers are known as unlike terms. Hence, if a polynomial has two variables, then all the same powers of any ONE variable will be known as like terms. Let us understand these two with the help of examples. For example, 2x and 3x are like terms. Whereas, 3y4 and 2x3 are unlike terms.
Degree of a Polynomial
The highest sum of the exponents is known as the degree of a polynomial. Let's consider an example. A polynomial 3x4 + 7 has a degree equal to four. Now, what would be the degree of the polynomial if there is more than one variable? Let's understand by taking the example of 3xy. In the above polynomial, the power of each variable x and y is 1. To calculate the degree in a polynomial with more than one variable, add the powers of all the variables in a term. So, we will get the degree of the given polynomial (3xy) as 2.
Standard Form of Polynomials
The standard form of a polynomial (polynomials in standard form) refers to writing a polynomial in the descending power of the variable. Let us understand this concept using an example. Let's write the polynomial 5+2x+x2 in the standard form. To show the above polynomial in standard form, we will first check the degree of the polynomial. In the given polynomial, the highest degree is 2. Next, we will check if there is a term with a degree less than 2, i.e., 1, and finally, if there is a term with degree 0, which is the constant term. 5+2x+x2 in standard form can be written as x2+2x+5. Always remember that in the standard form of a polynomial, the terms are written in decreasing order of the power of x.
3 Types of Polynomials
Polynomials can be categorized based on their degree and their power. Based on the numbers of terms, there are mainly three types of polynomials that are listed below:
Monomial is a type of polynomial with a single term. For example: x, −5xy, and 6y2. Binomial is a type of polynomial that has two terms. For example x+5, y2+5, and 3x3−7. While a Trinomial is a type of polynomial that has three terms. For example 3x3+8x−5, x+y+z, and 3x+y−5. However, based on the degree of the polynomial, polynomials can be classified into 4 major types:
- Zero or Constant polynomial
- Linear polynomial
- Quadratic polynomial
- Cubic polynomial
Polynomials with 0 degrees are called zero polynomials. For example, 3, 5, or 8. Polynomials with 1 as the degree of the polynomial are called linear polynomials. For example, x+y−4. Polynomials with 2 as the degree of the polynomial are called quadratic polynomials. For example, 2p2−7. Polynomials with 3 as the degree of the polynomial are called cubic polynomials. For example, 6m3 − mn + n2 − 4.
Addition and Subtraction of Polynomials
Addition and subtraction of polynomials are two basic operations that we use to increase or decrease the value of polynomials. Whether you wish to add numbers together or you wish to add polynomials, the basic rules remain the same. The only difference is that when you are adding 34 to 127, you align the appropriate place values and carry the operation out. However, when dealing with the addition and subtraction of polynomials, one needs to pair up like terms and then add them up. Otherwise, all the rules of addition and subtraction from numbers translate over to polynomials. Have a look at the image given here in order to understand how to add or subtract any two polynomials.
Important Notes on Polynomials
- Terms in a polynomial can be only separated by the '+' or '-' sign.
- For any expression to become a polynomial, the power of the variable should be a whole number.
- The addition and subtraction of a polynomial are possible between like terms only.
- All the numbers in the universe are called constant polynomials.
Given below is the list of topics that are closely connected to polynomials. These topics will also give you a glimpse of how such concepts are covered in Cuemath.
Solved Examples on Polynomials
Example 1: Mr. Stark wants to plant a few rose bushes on the borders of his triangular-shaped garden. If the sides of the garden are (4x−2) feet, (5x+3) feet, and (x+9) feet, what is the perimeter of the garden?
Perimeter of garden = (4x - 2) + (5x + 3) + (x + 9) = 4x + 5x + x - 2 + 3 + 9 = 10x + 10
∴ The perimeter is (10x+10) feet.
Example 2: The income of Mr. Smith is $ (2x2−4y2+3xy−5) and his expenditure is $ (−2y2+5x2+9). Use the concept of subtraction of polynomials to find his savings.
We all know that Savings = Income - expenditure. Now, applying the same thing here, we will get:
Savings = 2x2−4y2+3xy−5−(9−2y2+5x2) = 2x2−4y2+3xy−5+2y2−5x2−9 = −3x2−2y2+3xy−14
Hence, his saving will be $(−3x2−2y2+3xy−14).
FAQs on Polynomials
What are Monomials, Binomials, and Trinomials?
Monomial is a type of polynomial with a single term. For example x, −5xy, and 6y2. While a binomial will be having two terms. For example x+5, y2+5, and 3x3−7. While a Trinomial is a type of polynomial that has three terms. For example 3x3+8x−5, x+y+z, and 3x+y−5.
Is 8 a Polynomial?
8 is a Polynomial. It is known as a constant polynomial.
What is a Polynomial Equation?
A polynomial equation is when two different polynomials are combined together by the means of an equal-to sign. In this case, the expression then becomes a polynomial equation.
Why are Polynomials Important?
Polynomials form a large group of algebraic expressions. Any expression with only positive powers of the variables is termed as polynomials. As they cover such a huge chunk of all algebraic expressions, they tend to have a wide variety of applications.
How to Multiply and Divide Polynomials?
For the multiplication of polynomials, there are three laws that are to be kept in mind – distributive law, associative law, and commutative law. For division, the most common method used to divide one polynomial by another is the long division method.
Is Zero a Polynomial?
Number 0 is a special polynomial called "Constant Polynomial."