POLYNOMIALS

Mathematics

Name: Advetya A PillaiClass: X ‘B’

School : K.V Coimbatore

WHAT IS A POLYNOMIAL ?

A polynomial is an expression consisting of variables and coefficients , that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

if p(x) is a polynomial in x, the highest power of x in p(x) is called the degree of the polynomial p(x).

For example : 4x + 2, 25y – 25

Different types of polynomials

A polynomial of degree 1 is called a linear polynomial.

e.g.: 4x, 8y A polynomial of degree 2 called a quadratic

polynomial e.g.: -𝒙 8 A polynomial of degree 3 is called a cubic

polynomial. e.g.:𝒙,

zero of a polynomial

As p(–1) = 0 and p(4) = 0, –1 and 4 are called the zeroes of the quadratic polynomial x2 – 3x – 4. More generally, a real number k is said to be a zero of a polynomial p(x), if p(k) = 0

In general, if k is a zero of p(x) = ax + b,

then p(k) = ak + b =0, i.e., k=

So ,The zero of the linear polynomial a 𝒙+b is

=

Relationship between Zeroes and Coefficients of a Polynomial

Let α and β be the zeroes of the polynomial ax² + bx + c.

Sum of its zeroes = or

Product of its zeroes=

or =

Greek letters pronounced as “alpha” and “Beta” respectively

Numerical Problem1) Find the zeroes of the quadratic polynomial, and verify

the relationship between the zeroes and the coefficients.Ans: 𝒙𝟐+𝟕𝒙+𝟏𝟎Let , in eq.

a= 1 , b=7 , c=10 = = -7 ,

= = 10

Now, 1st zero : ( 2nd zero:(

Then ,

The sum of roots (= = Sum of products( =

Products of roots( =

a ,

Numerical Problem

Find a cubic polynomial whose zeros are -2 ,-3 and -1 .

Ans: Let,

1) 3)

Now,

There fore , is the required cubic polynomial

Division Algorithm for Polynomials

the division algorithm for polynomials states that “If f() and g() are two polynomials such that degree of f() is greater than degree of g() where g() ≠ 0, then there exists unique polynomials q() and r() such that f() = g().q() + r() where r() = 0 or degree of r() < degree of g().Hence by division rule it can written as, Dividend = divisor x quotient + remainder

This holds good even for polynomials too.

Divide the highest degree term of the dividend by the highest degree term of the divisor and obtain the remainder.

If the remainder is O or degree of remainder is less than divisor, then we cannot continue the division any furthur,if degree of remainder is equal to or more than divisor repeat the first step.

Numerical Problem If the two zeros of the polynomial are and . Find the 3rd

zero .

Let, , Therefore , )=0

Ans:

𝒙𝟐+𝟎 𝒙−𝟐𝟐 𝒙−𝟏

XTherefore, the third zero =

Divide the polynomial by the polynomial . Find the Quotient and remainder.

Ans: Let,

𝒙𝟐−𝟒 𝒙+𝟖

𝟏𝟎𝒙−𝟑𝟑

25

Therefore , Quotient = 2

Remainder=