Polynomial

An expression of the form a$_{0}$x$^{n}$ + a$_{1}$x$^{n - 1}$ + a$_{2}$x$^{n - 2}$ + a$_{3}$x$^{n - 3}$ + ..... + a$_{n}$ where a$_{0}$, a$_{1}$, a$_{2}$, a$_{3}$, ....., a$_{n}$ are given numbers (real or complex), n is a non-negative integer and x is a variable is called a polynomial in x.

a$_{0}$, a$_{1}$, a$_{2}$, a$_{3}$, etc., are called the coefficients of x$^{n}$, x$^{n - 1}$, x$^{n - 2}$, x$^{n - 3}$, etc., respectively.

a$_{0}$x$^{n}$, a$_{1}$x$^{n - 1}$, a$_{2}$x$^{n - 2}$, a$_{3}$x$^{n - 3}$, ....., a$_{n}$ are called the terms of the polynomial.

a$_{n}$ is called the constant term. Clearly, it is also the coefficient of x$^{0}$.

If a$_{0}$ ≠ 0, the polynomial is said to be of degree n and the term a$_{0}$x$^{n}$ is called the leading term.

The general form of a polynomial of degree 1 is a$_{0}$x + a$_{1}$where a$_{0}$ ≠ 0.

The general form of a polynomial of degree 2 is a$_{0}$x$^{2}$ + a$_{1}$x + a$_{2}$ where a$_{0}$ ≠ 0.

A non-zero constant a$_{0}$ itself is said to be a polynomial of degree 0 while a polynomial all of whose coefficients are zero is said to be a zero polynomial and is denoted by 0 and no degree is assigned to it.

Since a polynomial is an expression containing the variable x, it is denoted by f(x), p(x) or g(x) etc.

The value of a polynomial f(x) for x = a where a is real number or a complex number is denoted by f(a).

In particular, if the coefficients a$_{0}$, a$_{1}$, a$_{2}$, a$_{3}$, .... of a polynomial f(x) be all real numbers, the polynomial f(x) is said to be a real polynomial.

Examples of polynomial:

(i) 7x$^{2}$ + 5x - 3 is a polynomial in x of degree 2 or a quadratic polynomial in x.

(ii) 4x$^{3}$ + 9x$^{2}$ - 4x + 2 is a polynomial in x of degree 3 or a cubic polynomial in x.

(iii) 5 - 2x$^{\frac{5}{3}}$ + 9x$^{2}$ is an expression but not a polynomial, since it contains a term containing x$^{\frac{5}{3}}$ , where $\frac{5}{3}$ is not a non-negative integer. 

● Factorization

• Polynomial
• Polynomial Equation and its Roots
  
• Division Algorithm
  
• Remainder Theorem
  
• Problems on Remainder Theorem
  
• Factors of a Polynomial
  
• Worksheet on Remainder Theorem
  
• Factor Theorem
  
• Application of Factor Theorem

10th Grade Math

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