# NCERT Class 10 Mathematics Polynomial Expressions CBSE Board Sample Problems (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

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

**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

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

**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 Theorems**

If x-a is a factor of polynomial p (x) then or if is the factor the polynomial p (x)

**Geometric Meaning of the Zero՚s 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 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

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

**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