Polynomials Overview
The meaning of Polynomials is ‘many terms’, and it consists of coefficients and variables. These coefficients can be added, subtracted, or multiplied for various mathematical operations. While this chapter imparts knowledge about important terms like factoring polynomials, studying through these notes will help you learn the concept right.
You will be able to solve the exercise questions and answer them correctly once you have thoroughly read these notes. Now ensure that your academic performance gets better with these quality notes covering the intricacies of Polynomials.
Introducing Polynomials
Every polynomial is said to have a constant, a variable, and an exponent. It may have more than one terms and the number of terms determine the type of polynomial it is. For instance, take x2 + 5x + 3 as a polynomial expression. Clearly, it has 3 terms and hence can be called a trinomial. Monomial, binomial, etc. are few other kinds of polynomials here.
In case polynomials are classified depending upon their degree, they are segregated into –
Linear – Expressions having degree as 1.
Cubic – Expressions having degree as 3.
Quadratic – Expressions having degree as 2.
Examples of Polynomial
x + y
25
2x + y + 5
a + b + c + d
x2 + x + 2
x3 + y2 + 2x + 2
The algebraic expression for writing polynomials is as follows –
p (x) = a0xn + a1xn-1 + a2xn-2 + … an
Where, a0, a1, … … ... an denotes the real numbers and the value of n is a positive integer.
Factor Theorem
Consider a polynomial p (x) with degree equal to or greater than one, where ‘a’ is any real number. Then, we can conclude,
If p (a) is ‘0’, then (x – a) will be a factor of p (x).
If (x – a) factorises p (x), then p (a) will be 0.
Remainder Theorem
Consider a polynomial q (x) with degree equal to or greater than one, where ‘a’ is any real number. Then, we can conclude, dividing polynomials q (x) by a linear polynomial (x – a), then its remainder should be q (a).
Adding and Subtracting Polynomials
You can also add or subtract polynomials. To do so, you must add the like terms together or subtract from like terms.
For instance, take two polynomials, as shown below.
3 x2 + 5x + 8,
and 2 x2 – x – 2.
Place the like terms together and proceed to add.
3 x2 + 2 x2 + 5x – x + 8 – 2
Add the like terms together to get
(3 + 2) x2 + (5 – 1) x + (8 – 2)
5 x2 + 4 x + 6
Similarly, you can add or subtract polynomial terms by placing the like terms together and adding them.
In case of subtraction, consider these polynomials 3 x2 + 5x + 8 and 2 x2 – x – 2.
Place the like terms together and proceed to subtract.
3 x2 – 2 x2 + 5x + x + 8 + 2
Add the like terms together to get this
(3 – 2) x2 + (5 + 1) x + (8 + 2)
x2 + 6 x + 10
Now that you are familiar with the idea of multiplying polynomials, you will be able to solve the exercise questions effortlessly. It is critical to learn the theoretical concept and the method so that you can solve mathematical questions quickly.
The quality notes prepared by our expert tutors are meant to help you learn the concepts in an easy manner. Now start preparing for your upcoming exam with our notes and always score high grades in the exam. Now you can also download our Vedantu app for easier access to these materials.
FAQs on Polynomials
1. What is a polynomial in Maths?
A polynomial is an algebraic expression that consists of variables (or indeterminates), coefficients, and exponents, combined using addition, subtraction, and multiplication. The key rule is that the exponents of the variables must be non-negative integers (i.e., 0, 1, 2, 3, ...). An expression is not a polynomial if it contains division by a variable or if a variable has a negative or fractional exponent.
2. How can you identify if an algebraic expression is a polynomial?
To identify if an expression is a polynomial, check for the following conditions:
- Variable Exponents: The exponents of all variables must be whole numbers (0, 1, 2, ...). For example, x³ is valid, but x⁻² is not.
- No Variables in the Denominator: An expression like 1/x is not a polynomial because it is equivalent to x⁻¹.
- No Variables Under a Radical: An expression like √x is not a polynomial because it is equivalent to x¹/², which has a fractional exponent.
For example, 3x² + 2x - 5 is a polynomial, while 3/y + 2y is not.
3. What are the different types of polynomials based on their terms and degree?
Polynomials can be classified in two main ways:
- Based on the number of terms:
- Monomial: A polynomial with one term (e.g., 5x²).
- Binomial: A polynomial with two terms (e.g., 3x + 4).
- Trinomial: A polynomial with three terms (e.g., 2x² - 5x + 1).
- Based on the degree (highest exponent):
- Constant Polynomial: A polynomial of degree 0 (e.g., 7).
- Linear Polynomial: A polynomial of degree 1 (e.g., x - 2).
- Quadratic Polynomial: A polynomial of degree 2 (e.g., x² + 3x - 2).
- Cubic Polynomial: A polynomial of degree 3 (e.g., 4x³ - x).
4. How is the degree of a polynomial determined?
The degree of a polynomial is the highest exponent of the variable in the expression. For a polynomial with a single variable, you just find the largest exponent. For example, in 6x⁵ + 3x² - 10, the degree is 5. For polynomials with multiple variables in a single term, you add the exponents of the variables in that term. The highest sum is the degree of the polynomial. For example, in 7x²y³ + 4xy² + 9, the degree of the first term is 2+3=5, the second is 1+2=3. Therefore, the degree of the polynomial is 5.
5. Why is a constant number, like 7, considered a polynomial?
A constant number like 7 is considered a polynomial because it can be written in the standard polynomial form. We can express 7 as 7x⁰. According to the definition of a polynomial, the exponent of the variable must be a non-negative integer. Since 0 is a non-negative integer, the expression 7x⁰ is a valid polynomial. This specific type is known as a constant polynomial, and its degree is 0.
6. Are expressions with radicals or fractions always non-polynomials?
Not necessarily. The rule about radicals and fractions applies only to the variables, not the coefficients. An expression is not a polynomial if a variable is under a radical (e.g., √x) or has a fractional exponent. However, if the coefficients are radicals or fractions, the expression can still be a polynomial. For instance:
- √3x² + 5x is a polynomial because the radical (√3) is a coefficient, and the exponents of 'x' (2 and 1) are whole numbers.
- x² + √x is NOT a polynomial because the variable 'x' is under a radical sign.
7. What is the importance of finding the 'zeroes' of a polynomial?
The 'zeroes' of a polynomial are the values of the variable that make the polynomial's value equal to zero. Finding them is a fundamental concept in algebra for several reasons:
- Solving Equations: Finding the zeroes is equivalent to finding the roots or solutions of a polynomial equation P(x) = 0.
- Graphing: Geometrically, the real zeroes of a polynomial are the x-intercepts, which are the points where the graph of the polynomial crosses the x-axis.
- Factoring: As per the Factor Theorem, if 'a' is a zero of a polynomial P(x), then (x - a) is a factor of P(x). Finding zeroes helps in breaking down complex polynomials into simpler factors.
