## Polynomials

Algebraic expression containing many terms of the form ax^{n}, n being a non-negative integer is called a polynomial. i.e., f(x) = a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + …….. + a_{n-1}x^{n-1} + a_{n}x_{n}, where x is a variable, a_{0}, a_{1}, a_{2}, …… a_{n} are constants and a_{n} ≠ 0.

**Example:** 4x^{4} + 3x^{3} – 7x^{2} + 5x + 3, 3x^{3} + x^{2} – 3x + 5.

**(1) Real polynomial:**

f(x) = a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + …….. + a_{n}x_{n }is called real polynomial of real variable x with real coefficients.

**Example:** 3x^{3} – 4x^{2} + 5x – 4, x^{2} – 2x + 1 etc. are real polynomials.

**(2) Complex polynomial:**

f(x) = a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + …….. + a_{n}x_{n} is called complex polynomial of complex variable x with complex coefficients.

**Example:** 3x^{2} – (2+4i)x + (5i-4), x^{3} – 5ix^{2} + (1+2i)x + 4 etc. are complex polynomials.

**(3) Degree of polynomial:**

Highest power of variable x in a polynomial is called degree of polynomial.

**Example:** f(x) = a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + …….. + a_{n}x_{n} is a n degree polynomial.

f(x) = 4x^{3} + 3x^{2} – 7x + 5 is a 3 degree polynomial.

A polynomial of second degree is generally called a quadratic polynomial. Polynomials of degree 3 and 4 are known as cubic and biquadratic polynomials respectively.

**(4) Polynomial equation:**

If f(x) is a polynomial, real or complex, then f(x) = 0 is called a polynomial equation.

**Read More:**

- What is a Polynomial?
- Types of Polynomials
- Monomials, Binomials, and Polynomials
- Adding Polynomials
- Subtracting Polynomials
- Dividing Polynomials
- Polynomials – Long Division
- Degree (of an Expression)
- Special Binomial Products
- Multiplying Binomials
- Difference of Two Cubes
- Polynomial Remainder Theorem
- Factoring in Algebra
- Factorization of Polynomials Using Factor Theorem
- How do you use the factor theorem?
- How to factorise a polynomial by splitting the middle term?
- Review Factoring Polynomials
- Zeros of a Polynomial Function
- Factors and Coefficients of a Polynomial
- Roots of Polynomials: Sums and Products
- Solving Polynomial Equations of Higher Degree
- Examining Graphs of Polynomial Equations of Higher Degree