1. Different types of Polynomial Equations 2. Quadratic Equations

**Basics 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)= a*_{n}* xn *+ *a*_{n}_{-1} *xn*-1 +...+ *a*_{1} *x *+ *a*_{0} ……….(1)

where *a*_{r}* *∈ **C** are constants, *r
*= 0,1, 2,K, *n *with *a** _{n}* ≠ 0 . The variable

When all the coefficients of a polynomial *P *are real, we
say **“ P is a
polynomial over **

The function *P *defined by *P *( *x*) = *a*_{n}* x*^{n}* *+ *a*_{n}_{−1} *x*^{n}^{−1} +...+ *a*_{1}*x *+ *a*_{0} is called a **polynomial
function. **

The equation

*an xn *+ *a*_{n}_{-1} *x*^{n}^{-1} +...+ *a*_{1} *x *+ *a*_{0} = 0 ……….(2)

is called a **polynomial equation**.

If *a*_{n}*c*^{n}* *+ *a*_{n}_{−1}*c*^{n}^{−1} +...+ *a*_{1}*c *+ *a*_{0} = 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 *a*_{r}* *are called **coefficients. **The coefficient *a*_{n}* *is called the **leading coefficient **and the term *a*_{n}*x*^{n}* *is called the **leading term**. The coefficients may
be any number, real or complex. The only restriction we made is that the
leading coefficient *a*_{n}* *is nonzero. A polynomial with the leading
coefficient 1 is called a **monic polynomial***.*

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 6x^{2} + 7x^{2} y^{3} + 8z =9 are equations
in three variables x , y , z ; x^{2}- 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 *x*^{2} − 5*x *+ 6 and 3 is a root or
solution of the equation *x*^{2} − 5*x *+ 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 sin^{2} *x *+ cos^{2} *x *= 1 is an
identity on **R** , while sin *x *+ cos *x *= 1 and sin^{2} *x *+ cos^{2} *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 3*x*^{−}^{2} +1 and 5*x*^{1/2} +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 *ax*^{2} + *bx *+ *c *=0, *b*^{2} - 4*ac *is called the discriminant and it is usually denoted by
Δ.
We know that

are roots of the *ax*^{2} +
*bx *+ *c *= 0 . The two roots together are usually
written as

It is unnecessary to emphasize that *a* ≠ 0 , since by saying that *ax*^{2} + *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.

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12th Mathematics : UNIT 3 : Theory of Equations : Basics and types of Polynomial Equations |

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