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

- We have studied polynomial expression in one variable and their degrees in the previous classes.
- Now we know that the highest power of x in p(x) is called the degree of the polynomial p(x).A polynomial of degree 1 is called a linear polynomial , degree 2 is called quadratic polynomial,degree 3 is called a cubic polynomial.
- 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).A real number k is said to be a zero of a polynomial p(x), if p(k) = 0

- Check out Polynomial expression Degree,Value,Zeroes for the details on the above
- Polynomials Addition ,Subtraction ,Multiplication and division were also learned. Check out below links

Adding And subtracting Polynomials

Multiplying And Dividing Polynomials - Factoring of Polynomials can be done using grouping,split mid-term method,identity method. This we also learned in earliar classes.Check out How to factor polynomials for details
- This chapter we will focussing on finding the relationship between the coefficients and Zeroes of the polynomials expression. We will also study the division algorithm for 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 , shapes and zeroes 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. This can be seen in case of Quadratic polynomials
- If the degree n is odd, then one arm of the graph is up and one is down.This can be seen in case of cubic polynomials
- If the leading coefficient $a_n$ is positive, the right arm of the graph is up.
- If the leading coefficient $a_n$ is negative, the right arm of the graph is down

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

- Now that we know about the Geometrical meaning of the Zeros of the Polynomials, lets us the check the relation between the coefficient and zero's of the Polynomials.

- This are very useful formula and it can be used in many places.

- Class 10 Polynomial Syllabus includes relation for Linear ,Quadratic and Cubic Polynomials

For a quatric polynomial

$p(x) = ax^4 + bx^3 + cx^2 + dx + e$

If $\alpha,\ \beta, \ \gamma\ and\ \delta$ are the zeroes of the polynomial p(x)

Then below relationship exists

$\alpha+\ \beta+\ \gamma+\ \delta=\ \frac{-b}{a}$

$ \left(\alpha+\ \beta\right)\left(\gamma+\ \delta\right)+\ \left(\alpha\beta\right)+\ \left(\gamma\delta\right)=\ \frac{c}{a}$

$(\alpha+\ \beta)\ \gamma\delta+\left(\gamma+\ \delta\right)\alpha\beta=\ -\frac{d}{a}$

$ \alpha\beta\gamma\delta=\ \frac{e}{a}$

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

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

**Notes****NCERT Solutions****Assignments**

Class 10 Maths Class 10 Science