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Class 10 Maths notes for Polynomial





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Polynomial expression


A polynomial expression S(x) in one variable x is an algebraic expression in x term as
S(x)=anxn+a(n-1)x(n-1)+a(n-2)x(n-2)+.....+ax+a0
Where an,an-1,....,a,a0 are constant and real numbers and an is not equal to zero

Some important points to remember
1) an,an-1,...,a,a0 are called the coefficients for xn,xn-1 ,..,x,x0
2) n is called the degree of the polynomial
3) when an,an-1,....,a,a0 it is called zero polynomial
4) A constant polynomial is the polynomial with zero degree, it is a constant value polynomial
5) A polynomial of one item is called monomial, two items binomial and three items as trinomial
6) 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 a example of polynomial
S(x) =x2 +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
a) Linear polynomial has only one root
b) A zero polynomial has all the real number as roots
c) A constant polynomial has no zeros
d) 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 an 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) ,the remainder will be P(b/a)
4) for (ax+b) => a(x+b/a),the remainder will be P(-b/a)
5) for (b-ax)=> -a(x-b/a),the 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
Whereever it satisfies the factor theorem, we are good
In this particular 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 polynomial

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 polynomial

Quadratic Polynomial


Quadratic polynomial
Quadratic polynomial
Quadratic polynomial

Cubic Polynomial

Cubic polynomial

N degree Polynomial


N degree Polynomial
More facts about the geometric shape of the Polynomial
1. If the degree n of a polynomial is even, then the arms of the graph are either both up or both down.
2. If the degree n is odd, then one arm of the graph is up and one is down.
3. If the leading coefficient an is positive, the right arm of the graph is up.
4. 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 zero’s of the Polynomial

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 polynomial 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
Divide P(x) by q(x)
P(x)=x4 +x +1
q(x)=x+1

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

So q(x)=x3-x2+x
r(x)=1

So
x4 +x +1=(x+1)(x3-x2+x)+1

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