- Important Polynomials Definitions
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- Geometric Meaning of the Zero's of the polynomial
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- Relation between coefficient and zero's of the Polynomial
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- Formation of polynomial when the zeros are given
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- Division algorithm for Polynomial

- NCERT Solutions Exercise 2.1
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- NCERT Solutions Exercise 2.2
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- NCERT Solutions Exercise 2.3
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- NCERT Solutions Exercise 2.4

- Polynomials important questions
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- Polynomials Problem and Solutions
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- Polynomial worksheets
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- Polynomial questions
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$S(x)=a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+.....+a_1x+a_0$

Where $a_n,a_{n-1},....,a_1,a_0$ are constant and real numbers and an is not equal to zero

- a
_{n},a_{n-1},...,a,a_{0}are called the coefficients for x^{n},x^{n-1},..,x^{},x^{0} - 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

$S(x)=x^2 +1$

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)

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

$p(x)=g(x).q(x)+r(x)$

Notes

1) The degree of the reminder r(x) is always less then divisor g(x)

- for (x-a) then remainder P(a)

- for (x+a) => x -(-a), then remainder will be P(-a)
- for (ax-b) => a(x-b/a) , then remainder will be P(b/a)
- for (ax+b) => a(x+b/a),then remainder will be P(-b/a)
- for (b-ax)=> -a(x-b/a),then remainder will be P(b/a)

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)= x

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

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

x

In General Term,

S(x)=a

Look for the factors in a

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

- 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 points will help in roughly drawing the graph of any polynomial

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

- Arrange the term in decreasing order in both the polynomial
- Divide the highest degree term of the dividend by the highest degree term of the divisor to obtain the first term,
- Similar steps are followed till we get the reminder whose degree is less than of divisor

Divide P(x) by q(x)

$P(x)=x^4 +x +1$

$q(x)=x+1$

Following the step outlined above,here is the division

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

r(x)=1

So

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

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

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

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

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