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Polynomials Notes for Class 10 Maths




Important Polynomials Definitions


Polynomial expression

A polynomial expression S(x) in one variable x is an algebraic expression in x term as
$S(x)=a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+.....+a_1x+a_0$
Where $a_n,a_{n-1},....,a_1,a_0$ are constant and real numbers and an is not equal to zero

Some important points to remember
  • an,an-1,...,a,a0 are called the coefficients for xn,xn-1 ,..,x,x0
  • n is called the degree of the polynomial
  • when $a_n,a_{n-1},....,a_1,a_0$ are zero, it is called zero polynomial
  • A constant polynomial is the polynomial with zero degree, it is a constant value polynomial
  • A polynomial of one item is called monomial, two items binomial and three items as trinomial
  • A polynomial of one degree is called linear polynomial, two degree as quadratic polynomial and degree three as cubic polynomial

Value of the polynomial:

< Lets take an example of polynomial
$S(x)=x^2 +1$
Then
When we put the value of x=2,then
S(2)=4+1=5
The S(2) is the called the value of polynomial at x=2
In General terms, the value of polynomial at x=a is S(a)

Zero's or roots of the polynomial

It is a solution to the polynomial equation S(x)=0 i.e. a number "a" is said to be a zero of a polynomial if S(a) = 0.
If we draw the graph of S(x) =0, the values where the curve cuts the X-axis are called Zeros of the polynomial
  • Linear polynomial has only one root
  • A zero polynomial has all the real number as roots
  • A constant polynomial has no zeros
  • A zero of polynomial need not to be 0

Division Of Polynomial

When a polynomial p(x) is divided by the polynomial g(x), we get quotient q(x) and remainder r(x)

$p(x)=g(x).q(x)+r(x)$
Notes
1) The degree of the reminder r(x) is always less then divisor g(x)

Remainder Theorem's

If p(x) is a polynomial of degree greater than or equal to 1 and p(x) is divided by the expression (x-a),then the reminder will be p(a)

Important notes
  1. for (x-a) then remainder P(a)
  2. for (x+a) => x -(-a), then remainder will be P(-a)
  3. for (ax-b) => a(x-b/a) , then remainder will be P(b/a)
  4. for (ax+b) => a(x+b/a),then remainder will be P(-b/a)
  5. for (b-ax)=> -a(x-b/a),then remainder will be P(b/a)

Factor's Theorem's

If x-a is a factor of polynomial p(x) then p(a)=0 or if p(a) =0,x-a is the factor the polynomial p(x)

Factorising a Polynomial by Factor Theorem


We know by factor theorem if (x-a) is the factor of the polynomial ,then P(a)=0.
Suppose the Polynomial is the form
P(x)= x3 +6x2+11x+6

Step 1 We need to look at the constant 6 and factorise it
The factor of 6 will be 1,2,3
Now we can try the polynomial for all the values -3,-2,-1,1,2,3
Wherever it satisfies the factor theorem, we are good
In this case
P(-1)=P(-2)=P(-3)=0, we can write like this
Step 2 x3 +6x2+11x+6=K(x+1)(x+2)(x+3)

We can put any value of x in this identity and get the value of x
In this particular case putting x=0, we get K=1

So, the final identity becomes
x3 +6x2+11x+6=(x+1)(x+2)(x+3)

In General Term,

S(x)=anxn+a(n-1)x(n-1)+a(n-2)x(n-2)+....+ax+a0

Look for the factors in a0/an, Take both the positive and negative values and find out which suites your polynomial and then find the value of k



Geometric Meaning of the Zero's of the polynomials

Lets us assume
y= p(x) where p(x) is the polynomial of any form.
Now we can plot the equation y=p(x) on the Cartesian plane by taking various values of x and y obtained by putting the values. The plot or graph obtained can be of any shapes
The zero's of the polynomial are the points where the graph meet x axis in the Cartesian plane. If the graph does not meet x axis ,then the polynomial does not have any zero's.
Let us take some useful polynomial and shapes obtained on the Cartesian plane

Linear polynomial

Linear polynomials

Quadratic Polynomial


Open Upward and downward Parabola
Open Upward parabola intersecting at single point and Open upward Parabola having no zeroes
Open Downward Parabola having no zeroes

Cubic Polynomial

Cubic polynomials

N degree Polynomial


N degree Polynomials
More facts about the geometric shape of the Polynomials
  • If the degree n of a polynomial is even, then the arms of the graph are either both up or both down.
  • If the degree n is odd, then one arm of the graph is up and one is down.
  • If the leading coefficient an is positive, the right arm of the graph is up.
  • If the leading coefficient an is negative, the right arm of the graph is down
These above points can also be applied to all the example polynomial given up.
These points will help in roughly drawing the graph of any polynomial

Relation between coefficient and zero's of the Polynomial:

Relation between coefficient and zeroís of the Polynomials

Formation of polynomial when the zeros are given

Formation of polynomial when the zeros are given

Division algorithm for Polynomial

Let's p(x) and q(x) are any two polynomials with q(x) ≠ 0 ,then we can find polynomial s(x) and r(x) such that
P(x)=s(x) q(x) + r(x)
Where r(x) can be zero or degree of r(x) < degree of g(x)

Steps to divide a polynomial by another polynomial
  1. Arrange the term in decreasing order in both the polynomial
  2. Divide the highest degree term of the dividend by the highest degree term of the divisor to obtain the first term,
  3. Similar steps are followed till we get the reminder whose degree is less than of divisor

Example-1
Divide P(x) by q(x)
$P(x)=x^4 +x +1$
$q(x)=x+1$

Solution
Following the step outlined above,here is the division
Division algorithm for Polynomials

So $q(x)=x^3-x^2+x$
r(x)=1

So
$x^4 +x +1=(x+1)(x^3-x^2+x)+1$

Quiz Time

Question 1 What are zeros of the quadratic polynomial x2 + 3x + 2 ?
A. 3,-2
B. -3,2
C. 3,2
D. -3,-2
Question 2 If p, q, r are the zeroes of the cubic polynomial ax3 + bx2 + cx + d, then ?
A. p+q+r=-b/a
B. pqr=-a/d
C. pq+qr+pr=-c/d
D. none of these
Question 3 Given that two of the zeroes of the cubic polynomial ax3 + bx2+ cx + d are 0, the third zero is .
A. b/a
B. d/a
C. c/a
D. -b/a
Question 4 If the zeroes of the quadratic polynomial ax2+bx +c, where c ≠ 0, are equal then
A. c and a have same sign
B. c and b have same sign
C. c and a have opposite sign
D. c and b have opposite sign
Question 5 Which one is not the zero of polynomial 3x3 - 5x2 - 11x - 3 ?
A. 3
B. -1
C.-1/3
D. 4
Question 6 Write a polynomial function in standard form with zeros at 5, -4, and -3?
A. p(x)=x3 + 2x2 -23x+7
B. p(x)=x3 - 60x2 +2x-23
C. p(x)=x3 + 2x2 +23x-1
D. p(x)=x3 + 2x2 -23x-60

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Reference Books for class 10

Given below are the links of some of the reference books for class 10 math.

  1. Oswaal CBSE Question Bank Class 10 Hindi B, English Communication Science, Social Science & Maths (Set of 5 Books)
  2. Mathematics for Class 10 by R D Sharma
  3. Pearson IIT Foundation Maths Class 10
  4. Secondary School Mathematics for Class 10
  5. Xam Idea Complete Course Mathematics Class 10

You can use above books for extra knowledge and practicing different questions.


Class 10 Maths Class 10 Science

Practice Question

Question 1 What is $1 - \sqrt {3}$ ?
A) Non terminating repeating
B) Non terminating non repeating
C) Terminating
D) None of the above
Question 2 The volume of the largest right circular cone that can be cut out from a cube of edge 4.2 cm is?
A) 19.4 cm3
B) 12 cm3
C) 78.6 cm3
D) 58.2 cm3
Question 3 The sum of the first three terms of an AP is 33. If the product of the first and the third term exceeds the second term by 29, the AP is ?
A) 2 ,21,11
B) 1,10,19
C) -1 ,8,17
D) 2 ,11,20






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