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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

A polynomial expression S(x) in one variable x is an algebraic expression in x term as

S(x)=a

Where a

1) a

2) n is called the degree of the polynomial

3) when a

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

Lets take a example of polynomial

S(x) =x

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)

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

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)

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)

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)

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

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

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

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

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)=x^{4} +x +1

q(x)=x+1

Solution.

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

Divide P(x) by q(x)

P(x)=x

q(x)=x+1

Solution.

Following the step outlined above,here is the division

So q(x)=x

r(x)=1

So

x

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