Home | | Maths 12th Std | Basics and types of Polynomial Equations

# Basics and types of Polynomial Equations

1. Different types of Polynomial Equations 2. Quadratic Equations

Basics of Polynomial Equations

## 1. Different types of Polynomial Equations

We already know that, for any non–negative integer n , a polynomial of degree n in one variable

x is an expression given by

P ≡ P(x)= an xn + an-1 xn-1 +...+ a1 x + a0             ……….(1)

where ar C are constants, r = 0,1, 2,K, n with an ≠ 0 . The variable x is real or complex.

When all the coefficients of a polynomial P are real, we say P is a polynomial over R . Similarly we use terminologies like P is a polynomial over C , P is a polynomial over Q, and P is a polynomial over Z

The function P defined by P ( x) = an xn + an−1 xn−1 +...+ a1x + a0 is called a polynomial function.

The equation

an xn + an-1 xn-1 +...+ a1 x + a0 = 0              ……….(2)

is called a polynomial equation.

If ancn + an−1cn−1 +...+ a1c + a0 = 0 for some c C , then c is called a zero of the polynomial (1) and root or solution of the polynomial equation (2).

If c is a root of an equation in one variable x, we write it as“ x = c is a root”. The constants ar are called coefficients. The coefficient an is called the leading coefficient and the term anxn is called the leading term. The coefficients may be any number, real or complex. The only restriction we made is that the leading coefficient an is nonzero. A polynomial with the leading coefficient 1 is called a monic polynomial.

### Remark:

We note the following:

·               Polynomial functions are defined for all values of x .

·               Every nonzero constant is a polynomial of degree 0 .

·               The constant 0 is also a polynomial called the zero polynomial; its degree is not defined.

·               The degree of a polynomial is a nonnegative integer.

·               The zero polynomial is the only polynomial with leading coefficient 0 .

·               Polynomials of degree two are called quadratic polynomials.

·               Polynomials of degree three are called cubic polynomials.

·               Polynomial of degree four are called quartic polynomials.

It is customary to write polynomials in descending powers of x . That is, we write polynomials having the term of highest power (leading term) as the first term and the constant term as the last term.

For instance, 2x +3y +4z= 5 and 6x2 + 7x2 y3 + 8z =9 are equations in three variables x , y , z ; x2- 4x + 5 =0 is an equation in one variable x. In the earlier classes we have solved trigonometric equations, system of linear equations, and some polynomial equations.

We know that 3 is a zero of the polynomial x2 5x + 6 and 3 is a root or solution of the equation x2 5x + 6 = 0 . We note that cos x = sin x and cos x + sin x = 1 are also equations in one variable x.

However, cos x sin x and cos x + sin x 1 are not polynomials and hence cos x = sin x and cos x + sin x = 1 are not “polynomial equations”. We are going to consider only “polynomial equations” and equations which can be solved using polynomial equations in one variable.

We recall that sin2 x + cos2 x = 1 is an identity on R , while sin x + cos x = 1 and sin2 x + cos2 x = 1 are equations.

It is important to note that the coefficients of a polynomial can be real or complex numbers, but the exponents must be nonnegative integers. For instance, the expressions 3x2 +1 and 5x1/2 +1 are not polynomials. We already learnt about polynomials and polynomial equations, particularly about quadratic equations. In this section let us quickly recall them and see some more concepts.

For the quadratic equation ax2 + bx + c =0, b2 - 4ac is called the discriminant and it is usually denoted by Δ. We know that are roots of the ax2 + bx + c = 0 . The two roots together are usually written as It is unnecessary to emphasize that a0 , since by saying that ax2 + bx + c is a quadratic polynomial, it is implied that a 0.

• ∆ > 0 if, and only if, the roots are real and distinct

• ∆ < 0 if, and only if, the quadratic equation has no real roots.

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
12th Mathematics : UNIT 3 : Theory of Equations : Basics and types of Polynomial Equations |