# 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 |

Zeroâ€™s or roots of the polynomial | It is a solution to the polynomial equation S(x) =0 i.e. a number â€śaâ€ť 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 Theoremâ€™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) |

Factorâ€™s Theoremâ€™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

Letâ€™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 zeroâ€™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 zeroâ€™s | Quadratic polynomial | |

6 | where and and Example | Parabola It does not intersect the x-axis It has no zeroâ€™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 | Zeroâ€™s | Relationship between Zeroâ€™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 | Zeroâ€™s | Polynomial Formed |

Linear | k=a | (x-a) |

Quadratic | and | Or Or Sum of the zeroâ€™s)x +product of the zeroâ€™s |

Cubic | and |

### Division Algorithm for Polynomial

Letâ€™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