Properties of Polynomials (Theorems of Polynomials)
Various properties of the polynomials are:
For two polynomials P(x) and Q(x)
- degree (P ± Q) ≤ max(deg P, deg Q)
- deg(P⋅Q) = deg P + deg Q
For two given polynomials P(x) and Q(x) we always have unique polynomials Q (Quotient Polynomial) and R (Residue Polynomial) such that,
- P = R.Q + R
If a polynomial P(x) is divided by polynomial x – a, then P(a) = 0 is always true. This is also called Bezout’s Theorem or Factor Theorem.
If we a polynomial Q divides the polynomial P then zeros of polynomial Q are the zeros of polynomial P. For a polynomial of n degree we have n roots real or complex. Remainder of a polynomial f(x) when divided by (x – a) is f(a).
Polynomials – Definition, Standard Form, Types, Identities, Zeroes
Polynomials: In mathematics, polynomials are mathematical expressions consisting of indeterminates (also called variables) and coefficients, that involve only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. They are used in various fields of mathematics, astronomy, economics, etc. There are various examples of the polynomials such as 2x + 3, x2 + 4x + 5, etc.
In this article, we will learn about, Polynomials, Degrees of Polynomials, Examples of Polynomials, Zeros of Polynomials, Polynomial Equations, and others in detail.
Table of Content
- What are Polynomials?
- Polynomials Definition
- Polynomials Examples
- Characteristics of Polynomials
- Standard Form of a Polynomial
- Degree of a Polynomial
- Degree of Single Variable Polynomial
- Degree of a Multivariable Polynomial
- Terms in a Polynomial
- Types of Polynomials
- Properties of Polynomials (Theorems of Polynomials)
- Operations on Polynomials
- Addition of Polynomials
- Subtraction of Polynomials
- Multiplication of Polynomials
- Division of Polynomials
- Factorization of Polynomials
- Methods of Factorization of Polynomial
- Greatest Common Factor (GCF)
- Substitution Method
- Grouping Method
- Difference of Two Squares Identity
- Zeros of Polynomial
- How to Find Zeros of Polynomials?
- Linear Polynomial
- Quadratic Polynomial
- Cubic Polynomial
- Higher Degree Polynomial
- Polynomial Identities
- Polynomial Equations
- Solving Polynomials
- Polynomial Functions
- Polynomials Class 9 Extra Questions
- Polynomials Class 10 Extra Questions
- Practice Problems on Polynomial
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