Assignments for Class 9 Mathematics Polynomials have been developed for Standard 9 students based on the latest syllabus and textbooks applicable in CBSE, NCERT and KVS schools. Parents and students can download the full collection of class assignments for class 9 Mathematics Polynomials from our website as we have provided all topic wise assignments free in PDF format which can be downloaded easily. Students are recommended to do these assignments daily by taking printouts and going through the questions and answers for Grade 9 Mathematics Polynomials. You should try to do these test assignments on a daily basis so that you are able to understand the concepts and details of each chapter in your Mathematics Polynomials book and get good marks in class 9 exams.

**Key Concepts** for Class 9 Mathematics Polynomials

**Constants :** A symbol having a fixed numerical value is called a constant.**Example :** 7, 3, -2, 3/7, etc. are all constants.

**Variables : **A symbol which may be assigned different numerical values is known as variable.

**C = 2π r C – circumference of circle**

**Example :**r – radius of circle

**Where 2 & π are constants. while C and r are variable**

**Algebraic expressions : **A combination of constants and variables. Connected by some or all of the operations +, -, X and ÷ is known as algebraic expression**Example :** 4 + 9x – 5x^{2}y + 3/8, xy etc

**Terms : **The several parts of an algebraic expression separated by ‘+’ or ‘-‘ operations are called the terms of the expression**Example : **x^{2} + 2x^{2}y + 4xy^{2} + y^{3} + 7 is an algebraic expression containing 5 terms x^{2}, 3x^{3}y, – 4xy^{2}, y3 & 7

**Polynomials : **An algebraic expression in which the variables involved have only non-negative integral powers is called a polynomial.

(i) 5x^{3 }–^{ }4x^{2} – 6x – 3 is a polynomial in variable x.

(ii) 5 + 8x^{3/2} + 4x^{-2} is an expression but not a polynomial**Polynomials are denoted by** p (x), q (x) and r (x) etc.

**Coefficients : **In the polynomial x^{3} + 3x^{2} + 1, coefficient of x^{3}, x^{2}, x are 1,3,3 respectively and we also say that +1 is the constant term in it.

**Degree of a polynomial in one variable :** In case of a polynomial in one variable the highest power of the variable is called the degree of the polynomial.

Classification of polynomials on the basis of degree.

** degree Polynomial Example**1 Linear x + 1, 2x + 3 etc.

(a)

**2 Quadratic etc. ax**

**(b)**^{2}+ bx + c etc.

**3 Cubic etc. x**

**(c)**^{3}– 3x

^{2}+ 1 etc

**4 Biquadratic x**

**(d)**^{4}– 1

**Classification of polynomials on the basis of no. of terms**

** No. of terms Polynomial & Examples.**

(i) 1 Monomial – 5, 3x, 1/3, y etc.

(ii) 2 Binomial – (3 + 6x), (x – 5y) etc.

(iii) 3 Trinomial – 2x^{2} + 4x + 2 etc.

**Constant polynomial :** A polynomial containing one term only, consisting a constant term is called a constant polynomial the degree of non-zero constant polynomial is zero.

**Zero polynomial : **A polynomial consisting of one term, namely zero only is called a zero polynomial.

The degree of zero polynomial is not defined.

**Zeroes of a polynomial : **Let p (x) be a polynomial. If p (a) = 0, then we say that α is a zero of the polynomial of p(x).

Remark : Finding the zeroes of polynomial p(x) means solving the equation p(x) = 0.

**Remainder theorem : **Let f (x) be a polynomial of degree n ≥ 1 and let a be any real number. When f (x) is divided by (x – α) then the remainder is (α)

**Factor theorem :** Let f (x) be a polynomial of degree n > 1 and let a be any real number.

(i) if (α) = 0 then (x – α) is factor of f (x)

(ii) if (x – α) is a factor of f (x) then f (a) = 0

**Factor : **A polynomial p (x) is called factor of q (x), if p (x) divides q (x) exactly.

**Factorization : **To express a given polynomial as the product of polynomials each of degree less than that of the given polynomial such that no such a factor has a factor of lower degree, is called factorization.**Example :** x^{2 }– 16 = (x + 4) (x – 4)

**Methods of Factorization :**

Factorization by taking out the common factor

e.g.

36q^{3}b – 60a^{2}bc = 12a^{2} (3a – 5c)** Factorizing by grouping**

ab + bc + ax + cx = (ab + bc) + (ax + cx)

= (a + c) + x(a + c)

= (a + c) (b + x)

**Factorization of quadratic trinomials by middle term splitting method.**

x^{2} + bc + c = x^{2} + (p + q) x + pq

= ( x + p) ( x + q)

**Identity : **Identity is a equation (trigonometric, algebraic ) which is true for every value of variable.

**Some algebraic identities useful in factorization:**

(i) (x + y)^{2} = x^{2} + 2xy + y^{2}

(ii) (x + y)^{2} = x^{2} – 2xy + y^{2}

(iii) x^{2} – y^{2} = (x – y) (x +y)

(iv) (x +a) (x + b) = x^{2} + (a + b) x + ab

(v) (x + y + z)^{2} = x^{2} + y^{2} + z^{2} + 2xy + 2yz + 2zx

(vi) (x +y)^{3} = x^{3 }+ y^{3} + 3xy (x – y)

(vii) (x – y)^{3} = x^{3} – y^{3} – 3xy (x – y)

(viii) x^{3} + y^{3} + z^{3} – 3xyz = (x + y + z)(x^{2} + y^{2} + z^{2} – xy – yz -zx)

x^{3} + y^{3} + z^{3} = 3xyz if x + y + z = 0

#### Class 9 Mathematics Polynomials Assignments

#### Class 9 Mathematics Polynomials Important Questions and Answers

**Question. Write the degree of each of the following polynomial.**

**(i) 2x – √5**

**Ans.**

1

**(ii) 3 – x + x ^{2} – 6x^{3}**

**Ans.**

3

**(iii) 9**

**Ans.**

0

**(vi) 8x ^{4 }– 36x + 5x^{7}**

**Ans.**

4

**(v) x ^{9} – x^{5} + 3x^{10} + 8**

**Ans.**

9

**(vi) 2 – 3x ^{2}**

**Ans.**

2

**Question.****If x/y + y/x = -1 where x ≠ 0, y ≠ 0 then find the value of x ^{3} – y^{3}**

**Ans.**

0

**Question.** The zeroes of the polynomial**p(x) = 3x ^{2} – 1 are**

(a) 1/3

(b) 1/√3

(c) -1/√3

(d) 1/√3 and -1/√3

**Ans.**

1/√3 & -1/√3

**Question.** Find the zeros of the polynomials given below :

**(i) p (x) = x – 5**

**Ans.**

x = 5

**(ii) q (x) = x + 4**

**Ans.**

x = – 4

**(iii) h (x) = 6x – 1**

**Ans.**

x = 1/6

**(vi) p (x) = ax + b**

**Ans.**

x = – b/a

**(v) r (x) = x ^{2} + 3x**

**Ans.**

x = 0, x = – 3

**(vi) l (x) = x ^{2} + 2x + 1**

**Ans.**

x = – 1, – 1

**Question.** Which of the following expressions is polynomial?

(i) x^{5} – 2x^{3} + x + 7

(ii) y^{3 }– √3y

(iii) 5√z – 6

(iv) x – (1/x)

(v) x108 – 1

(vi) 2vx – 27

(vii) x – 2 + 2x – 1 + 3

**Ans.**

(i), (ii), (v)

**Question.****Find the remainder when f (x) = 12x ^{3} – 13x^{2} – 5x + 7 is divided by (3x + 2)?**

**Ans.**

remainder = 1

**Question.** Factorize :

**(i) 9x ^{2 }– 16y^{2}**

**Ans.**

(3x + 4y) (3x – 4y)

**(ii) x ^{3} – x**

**Ans.**

x (x + 1) (x – 1)

**Question.** Find the value of a for which (x – a )is a factor of the polynomial.

f(x) = x^{5 }– a^{2}x^{3} + 2x + a – 3

**Ans.**

a = 1

**Question.** Factorize following expressions.

**(i) x ^{2} + 9 x + 18**

**Ans.**

(x + 6) (x + 3)

**(ii) x ^{2} – 4x – 21**

**Ans.**

(x -7) (x + 3)

**(iii) x ^{2} – 9x + 18**

**Ans.**

(x – 6) (x – 3)

**(iv) x ^{2} – 19x + 78**

**Ans.**

(x – 6) (x – 13)

**Question.** Calculate (997)^{2} using algebraic identities

**Ans.**

994009

**Question.** Factorize:

**(i) a (a – 1) – b (b – 1)**

**Ans.**

(a-b) (a+b-1)

**Question.** Calculate 103 X 107 using algebraic identities

**Ans.**

11021

**Question.** Expand**(2a + 3b + 4c) ^{2}**

**Ans.**

4a^{2} + 9b^{2} + 16c^{2} + 12zb + 12bc + 16ac

**Question.** Factorize 4x^{2} + y^{2} + z^{2 }– 4xy – 2yz + 4xz

**Ans.**

(2x – y + z)^{2}

**Question.** Expand

**(i) (4a + 5b) ^{2} **

**Ans.**

64a^{3} + 125b^{3 }+ 240a^{2}b + 300qb^{2}

**(ii) (5x – 3y) ^{3}**

**Ans.**

125x^{3} – 27y^{3 }+ 225x^{2}y + 135xy^{2}

**Question.** Evaluate

**(i) (95) ^{3}**

## Ans.

857375

**(ii) (106) ^{3}**

**Ans.**

1191016

**Question.** Factorize the following expressions.

**(i) 5x ^{2} – 20 xy**

**Ans.**

5x (x – 4y)

**(ii) 5a (b + c) – 7b (b + c)**

**Ans.**

(b + c) (5a – 7b)

**(iii) x (x – y) ^{2} + 3x^{2}y (x – y)**

**Ans.**

x (x – y) (x – y)

**(iv) 6ab – b ^{2} + 12ac – 3bc**

**Ans.**

(b +2c) (6a – b)

**(v) x ^{2} + 1/x^{2} + 2 – 2x – (2/x)**

**Ans.**

**Question.** Degree of zero polynomial is

(a) 1

(b) 0

(c) not defined

(d) none of these

**Ans.**

(c) not defined

**Question.** Factorize**a ^{3} + 27b^{3} + 8c^{3 }– 18abc**

**Ans.**

(a + 3b + 2c) (a^{2} + 9b^{2} +4c^{2} – 3ab – 6bc – 2ac)

**Question.** Factorize (p – q)^{3} (q -r)^{3} (r – p)^{3}

**Ans.**

3(p – q) (q – r) (r – p)

**Question.** Find the product (3x – 5y – 4) (9x^{2} + 25y^{2} + 15xy + 12x – 20y + 16)

**Ans.**

27x^{3 }– 125y^{3} – 64 – 180xy.

**Question.** Factorize

(i) x^{3} + 64

**Ans.**

(x + 4) (x^{2} – 4x + 16)

(ii) 27x^{3 }+ 125y^{3}

**Ans.**

(3x + 5y) (9x^{2} – 15xy + 25y)^{2}

(iii) 8a^{3} – 27b^{3}

**Ans.**

(2a = 3b) (4a^{2} + 6ab + 9b^{2})

(iv) 1- 64a^{3}

**Ans.**

(1 – 4a) (1 + 4a + 16a^{2})

**Question.****Which of the following expression is a polynomial?**

**Ans.**

**Question.****If (x + 2) and (x – 1) are factors of (x ^{2} + 10x^{2} + mx + n) then find value of m & n?**

**Ans.**

m = 7, n = -18

** Question.** If if x + y + z = 9 and xy + yz + zx = 23 then find the value of (x

^{3}+ y

^{3}+ z

^{3}– 3xyz)?

**Ans**.

108

** Question.** For what value of k is the polynomial p (x) = 2x

^{3}– kx

^{2}+ 3x + 10 exactly divisible by (x + 2)?

(a) -1/3

(b) 1/3

(c) 3

(d) -3

**Ans.**

-3

**Question.****Find the value of (368) ^{2} – (368)^{2}**

**Ans.**

737

## Assignments for Class 9 Mathematics Polynomials as per CBSE NCERT pattern

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