NCERT Solutions for Class 9th Mathematics
Chapter 2 – POLYNOMIALS
(Complete Downloadable Chapter Solution PDF file is at the bottom of the page)
EXERCISE 2.1
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 + t√2 (iv) y + 2/y (v) x10 + y3 + t50
2. Write the coefficients of x2 in each of the following:
(i) 2 + x2 + x (ii) 2 – x2 + x3 (iii) (π/2)x2 + x (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) 5x3 + 4x2 + 7x (ii) 4 – y2 (iii) 5t – √7 (iv) 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 2.2
1. Find the value of the polynomial 5x – 4x2 + 3 at
(i) x = 0 (ii) x = –1 (iii) x = 2
2. 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)
3. Verify whether the following are zeroes of the polynomial, indicated against them.
(i) p(x) = 3x + 1, x = – 1/3 (ii) p(x) = 5x – π, x = 4/5
(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, x = – m/l
(vii) p(x) = 3x2 – 1, x = -1/√3, 2/√3 (viii) p(x) = 2x + 1, x = 1/2
4. 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 2.3
1. Find the remainder when x3 + 3x2 + 3x + 1 is divided by
(i) x + 1 (ii) x – 1/2 (iii) x (iv) x + π (v) 5 + 2x
2. Find the remainder when x3 – ax2 + 6x – a is divided by x – a.
3. Check whether 7 + 3x is a factor of 3x3 + 7x.
EXERCISE 2.4
1. 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) x3 – x2 – (2 + √2)x + √2
2. 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 – 4x2 + x + 6, g(x) = x – 3
3. Find the value of k, if x – 1 is a factor of p(x) in each of the following cases:
(i) p(x) = x2 + x + k (ii) p(x) = 2x2 + kx + √2
(iii) p(x) = kx2 – √2x + 1 (iv) p(x) = kx2 – 3x + k
4. Factorise :
(i) 12x2 – 7x + 1 (ii) 2x2 + 7x + 3
(iii) 6x2 + 5x – 6 (iv) 3x2 – x – 4
5. Factorise :
(i) x3 – 2x2 – x + 2 (ii) x3 – 3x2 – 9x – 5
(iii) x3 + 13x2 + 32x + 20 (iv) 2y3 + y2 – 2y – 1
EXERCISE 2.5
1. Use suitable identities to find the following products:
(i) (x + 4) (x + 10) (ii) (x + 8) (x – 10) (iii) (3x + 4) (3x – 5)
(iv) (y2 + 3/2) (y2 – 3/2) (v) (3 – 2x) (3 + 2x)
2. Evaluate the following products without multiplying directly:
(i) 103 × 107 (ii) 95 × 96 (iii) 104 × 96
3. Factorise the following using appropriate identities:
(i) 9x2 + 6xy + y2 (ii) 4y2 – 4y + 1 (iii) x2 – y2/100
4. Expand each of the following, using suitable identities:
(i) (x + 2y + 4z)2 (ii) (2x – y + z)2 (iii) (–2x + 3y + 2z)2
(iv) (3a – 7b – c)2 (v) (–2x + 5y – 3z)2 (vi) [a/4 – b/2 + 1]2
5. Factorise:
(i) 4x2 + 9y2 + 16z2 + 12xy – 24yz – 16xz
(ii) 2x2 + y2 + 8z2 – 2√2xy + 4√2yz – 8xz
6. Write the following cubes in expanded form:
(i) (2x + 1)3 (ii) (2a – 3b)3 (iii) [3x/2 + 1]3 (iv) [x – 2y/3]3
7. Evaluate the following using suitable identities:
(i) (99)3 (ii) (102)3 (iii) (998)3
8. Factorise each of the following:
(i) 8a3 + b3 + 12a2b + 6ab2 (ii) 8a3 – b3 – 12a2b + 6ab2
(iii) 27 – 125a3 – 135a + 225a2 (iv) 64a3 – 27b3 – 144a2b + 108ab2
(v) 27p3 – 1/216 – 9p2/2 + p/4
9. Verify : (i) x3 + y3 = (x + y) (x2 – xy + y2) (ii) x3 – y3 = (x – y) (x2 + xy + y2)
10. Factorise each of the following:
(i) 27y3 + 125z3 (ii) 64m3 – 343n3
[Hint : See Question 9.]
11. Factorise : 27x3 + y3 + z3 – 9xyz
12. Verify that x3 + y3 + z3 – 3xyz = (x + y + z) [(x – y)2 + (y – z)2 + (z – x)2]
13. If x + y + z = 0, show that x3 + y3 + z3 = 3xyz.
14. 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