In Mathematics, polynomials are termed as an expression that consists of variables, coefficients, the arithmetic operations of addition, subtraction, multiplication, and non-negative integer exponents. The word polynomial is derived from two words namely poly meaning “many” and nominal meaning “terms”. Therefore, polynomial means “many terms”.
x + 8, 4x² + 8x+ 2, -6, 5x³y² – 3y² + 2x – 1, 1 2 x² – 2 3x + 34
Polynomials play a significant role in the language of Mathematics and Algebra. Polynomials are widely used in almost every field of Mathematics. They are used to express numbers as a result of Mathematical operations. Polynomials also constitute a fundamental role in other types of Mathematical expression such as rational expression.
Many mathematical calculations that are performed in everyday life can be interpreted as polynomials. For example, summing the cost of confectionary items on a grocery bill, calculating the distance traveled by a vehicle or an object, or calculating the area, volume, perimeter of geometrical shapes are some of the mathematical calculations that can be interpreted as polynomials. Let’s learn this topic in a fun way at Cuemath.
What Are the Terms And Coefficient In Polynomials?
In the polynomial, 3x² + 4x, the expression 3x² and 4x are known as the terms of the polynomial. Similarly, we can say the polynomial 5x² + 3x + 2 has three terms namely 5x², 3x, and 2.
Each term of a polynomial has a coefficient in it. Hence, in the polynomial 8x³ + 6x²+ 5x – 3, the coefficient of x³is 8, the coefficient of x²is 6, the coefficient of x is 3, and the -3 is the coefficient of x⁰ as (x⁰ = 1).
Types of Polynomials on the Basis of the Number of Terms
Based on the number of terms, the polynomials are classified as:
Monomial: Polynomials having only one term are known as a monomial. For example, 2a, 3a²,3, -6a³, b, and y⁴.
Binomial: Polynomials having two terms are known as binomial. For example, 2a + 3, 3a²– a, b⁸⁰ + 1, etc.
Trinomial: Polynomials having three terms are known as trinomial. For example, a + a²+, b⁴ + b + 3, 3+ a – a², etc.
Degree of Polynomial
In a polynomial function, a degree is defined as the greatest exponent of that equation. The degree of a polynomial determines the several solutions that a function could have and several times a function will cut the x-axis when it is graphed.
Each polynomial function can have one or several terms. For example, the equation y= 3x15 + 7y2 has two terms i.e. 3x15 and 7y2. The degree for this polynomial equation is 17 as this is the highest degree of any term in a given polynomial equation.
Note: The degree of the non-zero constant polynomial is always equal to 0. For example, the degree of constant 7 is 0 as it can be written as 7xo.
Example 1: Find the degree of the polynomial x⁷ – x³ + 2.
The highest power of the exponent of a given polynomial equation is 7. Therefore, the degree of the polynomial x⁷ – x³ + 2 is 7
Example 2: Find the degree of the polynomial xy² + x²y³ – x³y¹.
To find the degree of the polynomial with more than one variable, you need to add the exponent of each variable in a term.
The degree of xy² = x¹y²= 1 + 2 = 3
The degree of x²y³ = 2 + 3 = 5
The degree of x³y¹ = 3 + 1 = 4
The highest degree is 5 as x²y³= 5
Therefore, the degree of the polynomial xy² + x²y³ – x³y¹is 5.
Types of Polynomials on the Basis of Its Degree
Constant Polynomial: A constant polynomial is termed as the polynomial of degree 0.
Linear Polynomial: A linear polynomial is termed as the polynomial of degree 1.
Quadratic Polynomial: A quadratic polynomial is termed as the polynomial of degree 2.
Cubic Polynomial: A cubic polynomial is termed as the polynomial of degree 3.