Important Formulas: Polynomials - Class 10 PDF Download

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

Where a_n, a_n-1,.......... a, a₀ are constant and real numbers and a_n is not equal to zero.

Some Important points to Note

1. a_n, a_n-1,a_n-2,...........a₁, a₀ are called the coefficients for xⁿ,x^n-1,...........,x¹_,x⁰
2. n is called the degree of the polynomial
3. when a_n ,a_n-1 ,a_n-2, ...... a₁,a₀ all are zero, 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

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Important Concepts on Polynomial 

1. 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 Zeroes of the polynomial
2. 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 remainder will be p(a)
3. 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)

Geometric Meaning of the Zeroes of the Polynomial: 
Let’s 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 purring the values. The plot or graph obtained can be of any shapes.
The zeroes 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 

Relation between coefficient and zeros of the polynomial: 

Formation of polynomial when the zeroes are given:

Division algorithm for polynomial:

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