CBSE Class 10 Maths Chapter 2 Polynomials

Class Notes of Ch 2 : Polynomials

Class 10th Maths

Polynomials

      Topics: 

• Polynomials
• Geometrical Meaning of Zeroes of Polynomials
• Relationship Zeroes & Coefficients of Polynomia
• Division Algorithm for Polynomials

Polynomials

The deal in the family is that, if Rosy gets x marks in her exam, her brother will give her 2x chocolates, her mother promised to give her x2 chocolates, her dad promised to give her x3 chocolates? How many chocolates will get?

Number of Chocolates Rosy will get, if she scored x marks is p(x) = 2x + x2 + x3

Number of Chocolates Rosy will get, if she scored 1 mark is p(1) = 2*1 + 12 + 13  = 4

Number of Chocolates Rosy will get, if she scored 2 mark is p(2) = 2*2 + 22 + 23  = 16

Number of Chocolates Rosy will get, if she scored 3 mark is p(3) = 2*3 + 32 + 33  = 42

The general expression p(x) = 2x + x2 + x3  is a polynomial.

Use of Polynomial in day to day life: We come across polynomial when we have to find area of a figure with variable dimensions. Also the formula to calculate compound interest on your money deposited in bank is a polynomial. It is also used in Physics, Chemistry & other field of science. Distance travelled by a ball on throw or bullet on fire can be calculated using a polynomial formula, S= ut + ½ at2

What we already know:

• Polynomials in one variable and their degrees. E.g. p(x)=3x2 +4x + 2
• If p(x) is a polynomial in x, highest power of x in p(x) is called degree of p(x). g. for polynomial p(x)= 3x2 +4x + 2, highest power is 2, & thus degree is 2.
• A polynomial of degree 1 is called a linear polynomial. E.g. p(x)=3x +4
• A polynomial of degree 2 is called a quadratic polynomial E.g. p(x)=3x2 +4
• A quadratic polynomial in x with real coefficients is of the form ax2 + bx + c, where a, b, are real numbers with a ≠ 0.
• A polynomial of degree 3 is called a cubic polynomial. E.g. p(x)=3x3 +4
• If p(x) is a polynomial in x, and if k is any real number, then the value obtained by replacing x by k in p(x), is called the value of p(x) at x = k, and is denoted by p(k).g. p(x)=3x +4 , p(2) = 3*2 + 4 = 10.
• A real number k is said to be a zero of a polynomial p(x), if p(k) = 0. g. p(x) =x3 -27, p(3)=0, thus 3 is said to be Zero of polynomial . p(x)=x3 -27 

Geometrical Meaning of Zeroes of Polynomials

A real number k is said to be a zero of a polynomial p(x), if p(k) = 0.

Polynomials can easily be represented graphically.

Zero of polynomial p(x) is x-coordinate of point where graph of p(x) intersects x-axis. Polynomial p(x) intersects the x-axis @ x=2, thus zero of this polynomial is 2.

Linear polynomial ax + b, a ≠ 0, has exactly one zero

E.g. Zero of linear polynomial p(x) = 2x -6   is 3 & thus the graph of this polynomial intersect x axis only once

Quadratic polynomial ax2 + bx +c, has nil, one or two zeroes

E.g. There are no Zeroes of Quadratic polynomial p(x) = x2 + 1 & thus the graph of this polynomial doesn’t intersect x axis.

E.g. There is one Zero of Quadratic polynomial p(x) = x2 -4x+4, that is 2 & thus the graph of this polynomial intersect x axis once at x=2.

E.g. There are Two Zeros of Quadratic polynomial p(x) = x2 – 4, that is +2, -2 & thus the graph of this polynomial intersect x axis at two place, x=2 & x=-2.

Cubic polynomial ax3 + bx2 +cx +d, has one, two or three zeroes. There can’t be any cubic polynomial with Nil zeroes.

E.g. There is one Zero of Cubic polynomial p(x) = x3, that is 0 & thus the graph of this polynomial intersect x axis once at x=0.

E.g. There are Two Zeros of Cubic polynomial p(x) = x3 – x2, that is 0 & 1, and thus the graph of this polynomial intersect x axis at two place, x=0 & x=1

E.g. There are three Zeroes of Cubic polynomial p(x) = x3 -4x, that is 0,+2 & -2, and thus the graph of this polynomial intersect x axis at three place, x=0, x=+2 & x=-2.

In general, given a polynomial p(x) of degree n, the graph of y = p(x) intersects the x- axis at at most n points. Therefore, a polynomial p(x) of degree n has at most n zeroes.

Also, Any polynomial of odd degree will never have nil zeroes.

Relationship between Zeroes and Coefficients of a Polynomial

We know that zero of a linear polynomial ax + b is –b/a

Let’s find the relationship between zeroes and coefficients of a quadratic polynomial.

For a quadratic polynomial p(x) = ax2 + bx + c, if the roots are α & β, then α + β = -b/a  & α * β = c/a

Numerical: Find the zeroes of the quadratic polynomial x2 + 7x + 10, and verify the relationship between the zeroes and the coefficients.

Solution:  p(x)=  x2 + 7x + 10= (x + 2)(x + 5)

p(x) = 0, when x= -2 or x= -5, therefore zeroes of p(x)=  x2 + 7x + 10 are -2 & -5.

Let’s see if α + β = -b/a  & α * β = c/a here a=1, b=7 & c =10

Sum of Zeroes = α + β = -2 + -5  = -7         -(i)

-b/a = -7/1  =  -7                                                -(ii)

Comparing i & ii,               α + β = -b/a 

Product of Zeroes = α * β   =  -2 *  -5   = 10            - (iii)

c/a = 10/1  = 10                                                                  - (iv)

Comparing iii & iv,            α * β = c/a

Numerical: Find a quadratic polynomial, the sum and product of whose zeroes are – 3 and 2, respectively.

Solution:   General form of quadratic polynomial is  p(x) = ax2 + bx + c   -(i)

α + β = -b/a  =  -3    or b = 3a                        -(ii)

& α * β = c/a =  2    or c = 2a                          -(iii)

Let a=k, then b=3k & c= 2k,    using (ii) & (iii)

Thus the equation will be  p(x) = kx2 + 3kx + 2k   or  p(x)  = k(x2 + 3x + 2)   where k is any real number.

Division Algorithm for Polynomials

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x).

Dividend = Divisor × Quotient + Remainder

Steps to divide Polynomials

1. Arrange terms of dividend & divisor in decreasing order of their degrees
2. Use Euclid formula to divide.

Also Read :

MATHS Revision Notes

Chapter:01  Real Numbers System
Chapter:02  Polynomials
Chapter:03  Pair of Linear Equations in Two Variables
Chapter:04  Quadratic Equation
Chapter:05  Arithemetic Progressions
Chapter:06  Triangles
Chapter:07  Coordinate Geometry
Chapter:08  Introduction to Trignometry
Chapter:09  Some Application Of Trignometry
Chapter:10  Circles
Chapter:11  Constructions
Chapter:12  Area Related to Cirles
Chapter:13  Surface Area Volume
Chapter:14  Stastistics
Chapter:15  Probability


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