__POLYNOMIALS__

__POLYNOMIALS__

**1**. An algebraic expression *p(x)* of the form *p(x)* = *a _{0} , + a_{1}x+a_{2} x^{2}*+ … +

*a*, where

_{n}x^{n}*a*are real numbers and all the indices of x are non-negative integers, is called a polynomial in x and the highest index n is called the degree of the polynomial, if

_{0}, a_{1}, a_{2}, …, a_{n}*a*,

_{n}_{}

0*a*+ … +

_{0}, + a_{1}x+a_{2}x^{2}*a*are called the terms of the polynomial and

_{n}x^{n}*a*an are called various coefficients of the polynomial

_{0}, a_{1}, a_{2}, …, a_{n}*f(x).*A polynomial in x is said to be in standard form when the terms are written either in increasing order or decreasing order of the indices of

*x*in various terms.

**2. Different types of polynomials**:

There are four polynomials based on degrees.

**(i) Linear polynomial:**

A polynomial of degree one is called a linear polynomial. It is of the form *ax + b*, where a,* b* € R and *a* _{} 0.

**(ii) Quadratic polynomial :**

A polynomial of degree two is called a quadratic polynomial. It is of the form *a*x^{2} + *bx* + *c*, where a, b, c € R and a _{} 0.

**(iii) Cubic polynomial:**

A polynomial of degree three is called a cubic polynomial. It is of the form *a*x^{3} +* b*x^{2}*+ cx + d*, where a, b, c, d € R and a _{}0.

**(iv) Biquadratic (or Quartic) polynomial:**

A polynomial of degree four is called a biquadratic (or quartic) polynomial. It is of the form *a*x^{4} + *b*x^{2}+ *c*x^{2} + dx + e = 0, where a, b, c, d, e € Rand a _{} 0.

**3. Value of the polynomial :**

If *p(x)* is a polynomial in *x*, and if a is any real constant, then the real number obtained by replacing *x* by a in *p(x),* is called the value of *p(x)* at a, and is denoted by *p(a).*

**4.Zero of a polynomial**:

A real number a is called a zero of a polynomial *p(x),* if p(a) =0.i.e., a zero of a polynomial is the value of the variable for which the value of the polynomial becomes zero.

**5**. The zeroes of a polynomial *p(x)* are precisely the *x*-coordinates of the points where the graph of *y = p(x)*intersects the x-axis.

**6**. A polynomial of degree *n *can have at the most *n* zeroes. So, a quadratic polynomial can have at the most 2 zeroes and a cubic polynomial can have at the most 3 zeroes.

**7**. If *a *and _{} are the zeroes of a quadratic polynomial *a*x^{2} + *bx* + *c*, then

a +_{} = _{}** , **a_{}** = **_{}** . **

**8**. If *a , _{} , *y are the zeroes of a cubic polynomial

*a*x

^{3}+

*b*x

^{2}

*+ cx + d*= 0 then

**9**. The division algorithm states that given any polynomial *p(x)* and any non-zero polynomial *g(x),* there are polynomials *q(x)* and *r(x)* such thatp*(x)* = *g(x) q(x) + r(x),* where *r(x) = 0* or degree *r(x)* <degree *q(x*).