Polynomials Definition Types Formula and Solved Examples
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 Complete Guide with Concepts and Applications
1. What is a polynomial in Maths?
A polynomial is an algebraic expression made up of variables, coefficients, and non-negative integer exponents combined using addition, subtraction, and multiplication. It has the general form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where n is a non-negative integer.
- Each part separated by + or − is called a term.
- The highest power of the variable is called the degree.
- Example: 3x² + 2x − 5 is a polynomial of degree 2.
2. What is the degree of a polynomial?
The degree of a polynomial is the highest exponent of the variable in the expression. It determines the polynomial’s behavior and graph shape.
- In 4x³ − 2x + 7, the degree is 3.
- In 5x² + x⁵ − 9, the degree is 5.
- A constant non-zero polynomial has degree 0.
3. How do you classify polynomials based on degree?
Polynomials are classified by degree using the highest power of the variable.
- Degree 0: Constant polynomial (e.g., 6)
- Degree 1: Linear polynomial (e.g., 2x + 3)
- Degree 2: Quadratic polynomial (e.g., x² − 4)
- Degree 3: Cubic polynomial (e.g., x³ + x)
- Degree 4 or more: Higher-degree polynomial
4. What is the standard form of a polynomial?
The standard form of a polynomial arranges terms in descending order of exponents. The general form is aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₀.
- Example: 3 + 2x² − x becomes 2x² − x + 3 in standard form.
- This form makes it easier to identify the degree and leading coefficient.
5. How do you add and subtract polynomials?
To add or subtract polynomials, combine like terms with the same variable and exponent.
- Step 1: Write expressions in standard form.
- Step 2: Combine like terms.
6. How do you multiply polynomials?
To multiply polynomials, multiply each term of one polynomial by each term of the other and combine like terms.
- Example: (x + 2)(x + 3)
- x(x + 3) + 2(x + 3)
- = x² + 3x + 2x + 6
- = x² + 5x + 6
7. What is the Remainder Theorem?
The Remainder Theorem states that when a polynomial f(x) is divided by (x − a), the remainder is f(a).
- Example: If f(x) = x² − 4 and divided by (x − 2),
- f(2) = 2² − 4 = 4 − 4 = 0.
8. How do you find the zeros of a polynomial?
The zeros of a polynomial are the values of x that make the polynomial equal to zero.
- Step 1: Set the polynomial equal to 0.
- Step 2: Factor or use formulas (like quadratic formula).
9. What is the difference between a monomial, binomial, and trinomial?
The difference lies in the number of terms in the polynomial.
- Monomial: One term (e.g., 5x²)
- Binomial: Two terms (e.g., x + 4)
- Trinomial: Three terms (e.g., x² + 3x + 2)
10. What are the real-life applications of polynomials?
Polynomials are used to model real-world problems involving growth, motion, and design.
- In physics, they model projectile motion.
- In economics, they represent cost and revenue functions.
- In engineering, they help design curves and structures.
