NCERT Class 10 Mathematics Formula CBSE Board Sample Problems Part 2
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Polynomial Expressions
A polynomial expression S(x) in one variable x is an algebraic expression in x term as
Where are constant and real numbers and is not equal to zero
Some Important Point to Note
S.no | Points |
1 | are called the coefficients for |
2 | n is called the degree of the polynomial |
3 | When 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 |
Important Concepts on Polynomial
Concept | Description |
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 |
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) |
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 putting 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
S.no | Graph obtained | Name of the graph | Name of the equation | |
1 | where a and b can be any values () Example |  Quadratic Polynomial Loading image••• | Straight line. It intersect the x axis at Example | Linear polynomial |
2 | where and and Example |  Quadratic Polynomial Loading image••• | Parabola It intersect the x axis at two points Example (3,0) and (4,0) | Quadratic polynomial |
3 | where and and Example |  Quadratic Polynomial Loading image••• | Parabola It intersect the x axis at two points Example (-2,0) and (4,0) | Quadratic polynomial |
4 | where and and Example |  Quadratic Polynomial Loading image••• | Parabola It intersect the x axis at one points | Quadratic polynomial |
5 | where and and Example |  Quadratic Polynomial Loading image••• | Parabola It does not intersect the x-axis It has no zero’s | Quadratic polynomial |
6 | where and and Example |  Quadratic Polynomial Loading image••• | Parabola It does not intersect the x-axis It has no zero’s | Quadratic polynomial |
7 | Where | It can be of any shape  Cubic Polynomial Loading image••• | It will cut the x-axis shape at the most 3 times | Cubic Polynomial |
8 | Where | It can be of any shape  Polynomial of N Degree Loading image••• | It will cut the x-axis shape at the most n times | Polynomial of n degree |
Relation between Coefficient and Zeroes of the Polynomial
S.no | Type of Polynomial | General form | Zero’s | Relationship between Zero’s and coefficients |
1 | Linear polynomial | 1 | ||
2 | Quadratic | 2 | ||
3 | Cubic | 3 | ||
4 |
Formation of Polynomial when the Zeroes Are Given
Types of polynomial | Zero’s | Polynomial Formed |
Linear | k=a | (x-a) |
Quadratic | and | Or Or Sum of the zero’s)x +product of the zero’s |
Cubic | and |
Division Algorithm for Polynomial
Let’s p(x) and q(x) are any two polynomial with ,then we can find polynomial s(x) and r(x) such that
Where r(x) can be zero or degree of degree of g(x)
Dividend