# NCERT Class 10 Mathematics Polynomial Expressions CBSE Board Sample Problems

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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 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

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 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 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 Zeroâ€™s 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 zeroâ€™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 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

**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