# NCERT Solutions for Class 9 Maths

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# Polynomials

Exercise 1

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

(i) 4x2 – 3x + 7      (ii) y2 + √2      (iii) 3√t + √2        (iv) (v) x10 + y3 + t50

2.    Write the coefficients of x2 in each of the following:

(i) 2+ x2 +x      (ii) 2 – x2+ x2        (iii) (iv) √2x – 1

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

4.    Write the degree of each of the following polynomials:

(i) p(x) = 5x3 + 4x2 + 7x       (ii) p(y) = 4 – y2     (iii) f (t) = 5t – √7     (iv) f (x) = 3

5.   Classify the following as linear, quadratic and cubic polynomials:

(i) x2 + x            (ii) x- x3              (iii) y + y2 + 4           (iv) 1 + x

(v) 3t                   (vi) r2                    (vii) 7x3

Exercise 2

6.   Find the value of the polynomial 5x – 4x2 + 3 at

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

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

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

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

8.    Verify whether the following are zeroes of the polynomial, indicated against them. 9.    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, d are real numbers.

Exercise 3

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

(i) x + 1         (ii) (iii) x          (iv) x + π          (v) 5 + 2x

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

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

Exercise 4

13.   Use suitable identities to find the following products: 14.   Evaluate the following products without multiplying directly:

(i) 103 × 107           (ii) 98 × 96                (iii) 104 × 96

15.   Factorize the following using appropriate identities: 16.  Factorize:

(i) 4x2 + 9y+ 16z+ 12xy – 24yz – 16xz

(ii) 2x2 + y2 + 8z2 – 2√2xy + 4√2yz – 8xz

17.  Write the following cubes in expanded form: 18.  Evaluate the following using suitable identities:

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

19.  Factorize each of the following: 20. Verify:

(i) x3 + y3 = (x+y)(x– xy + y2)         (ii) x – y = (x – y)(x2 + xy + y2)

21.  Factorize:

(i) 27y+ 125z3            (ii) 64m– 343n3

22.  Factorize: 27x3 + y3 + z3 – 9xyz

24.  If x+ y + z = 0, show that x+ y3 + z3 = 0.

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

(i) (-12)3 + (7)3 +(5)3            (ii) (28)3 +(-15)3 + (-13)3

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

(i) Area: 25a– 35a + 12        (ii) Area: 35y2 + 13y – 12

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

(i) Volume : 3x2 – 2x                (ii) Volume : 12ky2 + 8ky – 20k

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