Polynomials

Polynomials are mathematical expressions made up of variables and coefficients. Variables, which are also called indeterminates, are letters or symbols representing numbers. With polynomials, we can do math operations like adding, subtracting, and multiplying. We can also use positive whole numbers as exponents. However, we can’t divide by a variable.

For example, consider the polynomial: x² + x – 12. In this polynomial, there are three parts: x², x, and -12.

The word ‘polynomial’ comes from Greek words: ‘poly,’ which means ‘many,’ and ‘nominal,’ which means ‘terms.’ So, in simple terms, a polynomial means ‘many terms.’ Polynomials can have any number of terms, but they’re not infinite.

In this article, we’ll explore degrees, terms, types, properties, and polynomial functions to help you understand these concepts better.

What is a Polynomial?

A polynomial is a mathematical expression made up of variables (often represented by letters) and coefficients (numbers). These expressions are built by combining variables using arithmetic operations such as addition, subtraction, multiplication, and positive integer exponents. Importantly, polynomials do not involve division by variables.

Here’s a simple example of a polynomial:

3x² −5x+2

In this polynomial, we have:

• Variables: “x” (represented twice with different exponents).
• Coefficients: 3, -5, and 2.
• Arithmetic operations: Addition and subtraction.

Degree of a Polynomial

The degree of a polynomial is a fundamental concept in algebra that tells you the highest power of the variable (or indeterminate) in the polynomial expression. It helps classify and understand polynomials based on their complexity and behavior.

Types of Polynomials

Following are the types of polynomials mentioned below:

• Monomial
• Binomial
• Trinomial

Monomial

A monomial is a type of algebraic expression that consists of a single term. In other words, it’s a mathematical expression with just one part, typically made up of a coefficient (a numerical factor) and a variable raised to a non-negative integer exponent. Monomials can also be constants (numbers without variables).

Here are some examples of monomials:

1. Constant Monomial: A monomial can be a constant, which is simply a number. For instance:

• 5
• -3.14
• 0.25

2. Monomial with a Single Variable:

• 3x
• -2y²
• 0.5a³

3. Monomial with Multiple Variables: Monomials can also have more than one variable term, as long as each term has a non-negative integer exponent:

• 2xy
• 4a²b³c
• -0.75x²y⁴z²

4. Constant times a Variable: When a number is multiplied by a variable, it’s still a monomial:

• 6p
• -1.5q
• 2.5x³

Binomial

A binomial is a specific type of polynomial that consists of two terms. These two terms are usually connected by either addition or subtraction. Binomials are fundamental expressions in algebra and are often used to represent various mathematical and real-world relationships. The general form of a binomial is:

ax ⁿ+ bx ^m

Here, “a” and “b” are coefficients (numbers), “x” is the variable, and “n” and “m” are exponents (whole numbers). The exponents can be different, making one term have a higher degree (power) than the other.

Here are some examples of binomials:

2x+3: This is a binomial with two terms, where “2x” and “3” are the terms connected by addition. The coefficients are “2” and “3,” the variable is “x,” and the exponents are both implicitly “1.”

Trinomial

A trinomial is a polynomial with three terms. Each term consists of a variable raised to a certain exponent multiplied by a coefficient. Trinomials are a specific type of polynomial that is characterized by this three-term structure. They are often encountered in algebra and mathematics.

Here’s a general form of a trinomial:

ax ²+bx+c

Where:

• “a,” “b,” and “c” are coefficients, which can be any real numbers or constants.
• “x” is the variable, and it’s raised to the power of 2 in the first term, raised to the power of 1 in the second term, and not raised to any power (power of 0, which is 1) in the third term.

Here’s an example of a trinomial: 3x ² −5x+2

Polynomial Properties

Some of the important properties of polynomials are as follows:

• Property 1: Division Algorithm
• Property 2: Bezout’s Theorem
• Property 3: Remainder Theorem
• Property 4: Factor Theorem
• Property 5: Intermediate Value Theorem
• Property 6: The addition, subtraction and multiplication of polynomials
• Property 7: If a polynomial a is divisible by a polynomial b, then every zero of b is also a zero of a.
• Property 8: If a polynomial P is divisible by two co-prime polynomials a and b, then it is divisible by (a • b).
• Property 9: Descartes’ Rule of Sign
• Property 10: Fundamental Theorem of Algebra

Polynomial Operations

Polynomial operations involve various mathematical operations that you can perform on polynomials, which are algebraic expressions consisting of variables and coefficients. The primary polynomial operations include addition, subtraction, multiplication, division (in some cases), and evaluating polynomials. Here’s an overview of each operation:

1. Addition and Subtraction
2. Multiplication
3. Division (Long Division)

FAQs on Polynomials

What is a Polynomial?

A polynomial is a mathematical expression made up of variables, coefficients, and non-negative integer exponents. It consists of one or more terms, where each term is a product of a coefficient and a variable raised to a power.

What are polynomials Class 9?

In Class 9 mathematics, students typically learn about polynomials. They study the basic concepts of polynomials, including their definitions, operations (addition, subtraction, multiplication), and factorization.

What is a polynomial in math?

In mathematics, a polynomial is an algebraic expression consisting of variables, coefficients, and exponents. These expressions can be added, subtracted, multiplied, and evaluated for specific values of the variables.

Is zero a polynomial or not?

No, zero is not considered a polynomial. Polynomials must have non-negative integer exponents for their variables. A polynomial can't have a variable raised to the power of zero.

Is root 2 a polynomial?

No, √2 (square root of 2) is not a polynomial. Polynomials involve variables with non-negative integer exponents, and √2 is not a polynomial because it includes a square root.

Can 5 be a polynomial?

Yes, the constant value 5 can be considered a polynomial. It's a specific type of polynomial known as a constant polynomial because it has no variables; it's simply a constant term.

Why is 7 a polynomial?

The number 7 by itself is not a polynomial. A polynomial is an algebraic expression with variables and coefficients. However, 7 can be considered a constant polynomial because it's a constant term and fits the definition of a polynomial.