# NCERT Solutions for Class 9 Maths Unit 2

## Polynomials Class 9

### Unit 2 Polynomials Exercise 2.1, 2.2, 2.3, 2.4, 2.5 Solutions

In mathematics, a polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial of a single indeterminate (or variable), x, is x2 - 4x + 7, which is a quadratic polynomial.

Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra and algebraic geometry.

### Exercise 2.1 : Solutions of Questions on Page Number : 32

Q1 :

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.

(i) (ii) (iii)

(iv) (v)

(i)

Yes, this expression is a polynomial in one variable x.

(ii)

Yes, this expression is a polynomial in one variable y.

(iii)

No. It can be observed that the exponent of variable t in term is , which is not a whole number. Therefore, this expression is not a polynomial.

(iv)

No. It can be observed that the exponent of variable y in termis - 1, which is not a whole number. Therefore, this expression is not a polynomial.

(v)

No. It can be observed that this expression is a polynomial in 3 variables x, y, and t. Therefore, it is not a polynomial in one variable.

Q2 :

Write the coefficients of in each of the following:

(i) (ii)

(iii) (iv)

(i)

In the above expression, the coefficient of is 1.

(ii)

In the above expression, the coefficient of is - 1.

(iii)

In the above expression, the coefficient of is.

(iv)

In the above expression, the coefficient of is 0.

Q3 :

Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Q4 :

Write the degree of each of the following polynomials:

(i) (ii)

(iii) (iv) 3

Q5 :

Classify the following as linear, quadratic and cubic polynomial:

(i) (ii) (iii) (iv) (v)

(vi) (vii)

### Exercise 2.2 : Solutions of Questions on Page Number : 34

Q1 :

Find the value of the polynomial at

(i) x = 0 (ii) x = - 1 (iii) x = 2

Q2 :

Find p(0), p(1) and p(2) for each of the following polynomials:

(i) p(y) = y2 - y + 1 (ii) p(t) = 2 + t + 2t2 - t3

(iii) p(x) = x3 (iv) p(x) = (x - 1) (x + 1)

Q3 :

Verify whether the following are zeroes of the polynomial, indicated against them.

(i) (ii)

(iii) p(x) = x2 - 1, x = 1, - 1 (iv) p(x) = (x + 1) (x - 2), x = - 1, 2

(v) p(x) = x2 , x = 0 (vi) p(x) = lx + m

(vii) (viii)

Q4 :

Find the zero of the polynomial in each of the following cases:

(i) p(x) = x + 5 (ii) p(x) = x - 5 (iii) p(x) = 2x + 5

(iv) p(x) = 3x - 2 (v) p(x) = 3x (vi) p(x) = ax, a ≠ 0

(vii) p(x) = cx + d, c ≠ 0, c, are real numbers.

### Exercise 2.3 : Solutions of Questions on Page Number : 40

Q1 :

Find the remainder when x3 + 3x2 + 3x + 1 is divided by

(i) x + 1 (ii) (iii) x

(iv) x + π (v) 5 + 2x

Q2 :

Find the remainder when x3 - ax2 + 6x - a is divided by x - a.

Q3 :

Check whether 7 + 3x is a factor of 3x3 + 7x.

### Exercise 2.4 : Solutions of Questions on Page Number : 43

Q1 :

Determine which of the following polynomials has (x + 1) a factor:

(i) x3 + x2 + x + 1 (ii) x4 + x3 + x2 + x + 1

(iii) x4 + 3x3 + 3x2 + x + 1 (iv)

Q2 :

Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:

(i) p(x) = 2x3 + x2 - 2x - 1, g(x) = x + 1

(ii) p(x) = x3 + 3x2 + 3x + 1, g(x) = x + 2

(iii) p(x) = x3 - 4 x2 + x + 6, g(x) = x - 3

### Exercise 2.5 : Solutions of Questions on Page Number : 48

Q1 :

Use suitable identities to find the following products:

(i) (ii)

(iii) (iv)

(v)

Q2 :

Evaluate the following products without multiplying directly:

(i) 103 x 107 (ii) 95 x 96 (iii) 104 x 96

Q3 :

Factorise the following using appropriate identities:

(i) 9x2 + 6xy + y2

(ii)

(iii)

Q4 :

Expand each of the following, using suitable identities:

(i) (ii)

(iii) (iv)

(v) (vi)

Q5 :

Factorise:

(i)

(ii)

Q6 :

Write the following cubes in expanded form:

(i) (ii)

(iii) (iv)

Q7 :

Evaluate the following using suitable identities:

(i) (99)3 (ii) (102)3 (iii) (998)3

Q8 :

Factorise each of the following:

(i) (ii)

(iii) (iv)

(v)

Q9 :

Verify:

(i)

(ii)

Q10 :

Factorise each of the following:

(i)

(ii)

[Hint: See question 9.]

Q11 :

Factorise:

Q12 :

Verify that

Q13 :

If x + y + z = 0, show that .

Q14 :

Without actually calculating the cubes, find the value of each of the following:

(i)

(ii)

Q15 :

Give possible expressions for the length and breadth of each of thefollowing rectangles, in which their areas are given:

Q16 :

What are the possible expressions for the dimensions of the cuboids whose volumes are given below?