# Polynomials

Assume that*a*

_{1},

*a*

_{2},

*a*

_{3},

*a*

_{4},â€¦are real numbers and

*x*is a real variable. Then

*f*(

*x*) =

*a*

_{1}

*x*+

^{n}*a*

_{2}

*x*

^{n}^{-}

^{1}+

*a*

_{3}

*x*

^{n}^{-2}+â€¦ +

*a*

_{n}_{-}

_{1}

*x*+

*a*is called a polynomial.

_{n}x*x*

^{5}+ 3

*x*

^{4}+â€¦+

*x*is a polynomial in

*x*, where

*x*is a real variable.

# Degree of a Polynomial

*f*(

*x*) =

*a*

_{1}

*x*+

^{n}*a*

_{2}

*x*

^{n}^{-}

^{1}+

*a*

_{3}

*x*

^{n}^{-}

^{2}+â€¦+

*a*

_{n}_{-}

_{1}

*x*+

*a*is a polynomial of degree

_{n}x*n*, where

*a*

_{1}â‰ 0.

*x*

^{5}+ 3

*x*

^{4}+â€¦+

*x*is a polynomial of degree 5.

2

*x*

^{4}+

*x*

^{3}+ 4

*x*

^{2}+ 2

*x*+ 10 is a polynomial of degree 4.

4

*x*

^{3}+ 4

*x*

^{2}+ 2

*x*+ 10 is a polynomial of degree 3.

4

*x*

^{2}+ 2

*x*+ 10 is a polynomial of degree 2.

Remember that

- The degree of a polynomial is defined for real as well as complex polynomials.
- The degree of a polynomial cannot be a fraction.

# Polynomial Equation

If*f*(

*x*) is a real or complex polynomial, then

*f*(

*x*) = 0 is the corresponding equation. Simply put, when we equate any polynomial with zero, then it becomes equation.

*x*

^{5}+ 3

*x*

^{4}+â€¦+

*x*= 0 is an equation.

2

*x*

^{4}+

*x*

^{3}+ 4

*x*

^{2}+ 2

*x*+ 10 = 0 is an equation.

# Polynomial Inequation

If*f*(

*x*) is a real or complex polynomial, then

*f*(

*x*) â‰ 0 is the corresponding inequation. Simply put, if any polynomial is not equal to zero, i.e., either greater than zero or less than zero, then it is known as an inequation.

*x*

^{4}+

*x*

^{3}+ 4

*x*

^{2}+ 2

*x*+ 10 >0 is an inequation.

4

*x*

^{2}+ 2

*x*+ 10 < 0 is an inequation.

# Roots of an Equation

All the values of a variable satisfying the equation are known as the roots of the equation. It is known that an equation of degree

*n*will have*n*roots, real or imaginary. Roots can be the same or distinct. When the roots are the same, they are known as repeated roots.

# Quadratic Equation

Any equation of degree 2 is known as a quadratic equation.

*ax*^{2}+*bx*+*c*= 0 is known to be the standard equation of quadratic equation. This equation will have two roots either real or imaginary.

# Geometrical Meaning of Roots

For any given equation
For example,

For any quadratic equation

*y*=*f*(*x*) = 0, the number of times the graph of this equation cuts X-axis is equal to the distinct real roots of this equation.*y*= (*x*â€“1) (*x*+2) (*x*â€“2)=0 will intersect X-axis at three distinct points, namely*x*= 1, âˆ’2 and 2.For any quadratic equation

*f*(*x*) = (*x*â€“1) (*x*âˆ’3), the graph of this equation (as can be seen below) will intersect X-axis at two distinct points, namely*x*= 1 and*x*= 3.So, the equation

*f*(

*x*) =

*x*

^{2}â€“ 4

*x*+ 3 = 0 will have two roots, i.e., two values of

*x*= 1 and

*x*=3, which will satisfy this equation.

*p*â€™ and â€˜

*q*â€™ of the quadratic equation

*f*(

*x*) =

*ax*

^{2}+

*bx*+

*c*= 0 are such that

*f*(2) Ã—

*f*(âˆ’2) >0, then either both the roots â€˜

*p*â€™ and â€˜

*q*â€™ of this equation will lie inside â€“2 and 2 or both the roots â€˜

*p*â€™ and â€˜

*q*â€™ will lie outside the range of â€“2 and 2.

*p*â€™ and â€˜

*q*â€™ of the quadratic equation

*f*(

*x*) =

*ax*

^{2}+

*bx*+

*c*= 0 are such that

*f*(2) Ã—

*f*(âˆ’2) <0, then one of the two roots of this equation will lie inside this range and another will lie outside the range of â€“2 and 2.