NCERT Class 10 Mathematics Formula CBSE Board Sample Problems Part 2 (For CBSE, ICSE, IAS, NET, NRA 2022)
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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 |  | Straight line. It intersect the x axis at Example | Linear polynomial |
| 2 | where and and Example |  | Parabola It intersect the x axis at two points Example (3,0) and (4,0) | Quadratic polynomial |
| 3 | where and and Example |  | Parabola It intersect the x axis at two points Example (-2,0) and (4,0) | Quadratic polynomial |
| 4 | where and and Example |  | Parabola It intersect the x axis at one points | Quadratic polynomial |
| 5 | where and and Example |  | Parabola It does not intersect the x-axis It has no zero՚s | Quadratic polynomial |
| 6 | where and and Example |  | Parabola It does not intersect the x-axis It has no zero՚s | Quadratic polynomial |
| 7 | Where | It can be of any shape | It will cut the x-axis shape at the most 3 times | Cubic Polynomial |
| 8 | Where | It can be of any shape | 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