Solution of Chapter 2. Polynomials (Maths Important Questions Book)

Important Questions

1 Find the remainder when y3 + y2 - 2y + 5 is divided by y - 5.
2 Determine the remainder when polynomial p(x) is divided by x – 2.
3 Question-3 Show that x + a is a factor of xn +an for any odd +ve integer n.
4 When f(x) = x4 - 2x3 + 3x2 - ax is divided by x + 1 and x - 1, we get remainders as 19 and 5 respectively.
Find the remainder if f(x) is divided by x - 3.
5 What must be subtracted from 4x4 - 2x3 - 6x2 + x - 5 so that the result is exactly divisible by 2x2 + 3x - 2 ?
6 If both (x + 1) and (x - 1) are factors of ax3 + x2 - 2x + b, find a and b.
7 If x2 - 1 is a factor of ax4 + bx3 + cx2 + dx + e, show that a + c + e = b + d = 0.
8 Factorise each of the following expressions:
(i) 48x3 - 36x2
(ii) 5x2 - 15xy
(iii) 15x3y2z - 25xy2z3
9 Factorise
(i) 2x2(x + y) - 3(x + y) 
(ii) 5xy(5x + y) - 5y(5x + y)
(iii) x(x2 + y2 - z2) + y(x2 + y2 - z2) + z(x2 + y2- z2)
(iv) ab(a2 + b2 - c2) + bc(a2 + b2 - c2) + ca(a2+b2-c2)
10 Factorise
(i) 6ab - b2 + 12ac - 2bc
(ii) x2 + y - xy - x
(iii) x3 - 2x2y + 3xy2 - 6y3
(iv) a(a + b - c) – bc
11 Factorise each of the following expressions.
(i) 25x2y2 - 20xy2z + 4y2z2
(ii) 4x2 - 4x + 7
(iii) 
(iv) 4a2 + 12ab + 9b2 - 8a - 12b
12 Factorise
(i) 25x2 - 36y2
(ii) 2ab - a2 - b2 + 1
(iii) 36a2 - 12a + 1 - 25b2
(iv) a4 - 81b4
(v) a12b4 - a4b12
(vi) 4x2 - 9y2 - 2x - 3y  
13 Factorise by completing the squares.
(i) a4 + a2 + 1
(ii) y4 + 5y2 + 9
(iii) x4 + 4
(iv) x4 + 4x2 + 3
14 Factorise the following expressions:
(i) a3 - 27
(ii) 1 - 27x3
(iii) 8x3 - (2x - 3y)3
(iv) a8 - a2b6
(v) a3 - 5b3
15 Factorise the following
(i) 16p3q3 + 54r3
(ii) + 8b3
(iii) 2 a3 + 3 b3
(iv) 8a4b + ab4
(v) a7 - 64a
16 Factorise
(i) x3 + 9x2+ 27x + 27
(ii) x3 – 9x2y + 27xy2 – 27y3
17 Using identities, find the value of 
(i) 1012
(ii) 982
(iii) (0.98)2 
(iv) 11 × 9 
(v) 190 × 190 - 10 × 10
18 If x + y = 5 and xy = 4, find x - y, using identities.
19 Expand
(i) (x + 5y + 6z)2
(ii) (2a - 3b + 4c)2
(iii) (-a + 6b + 5c)2                 
(iv) (-p +4q - 3r)2
20 Expand
(i) (2x + 5y)3
(ii) (5p – 3q)3
(iii) (-a + 2b)3 
21 Evaluate, using identities
(i) 1023
(ii) 993
22 Simplify (2a + b)3 + (2a- b)3
23 Evaluate, using the identities
(i) 505 x 503,
(ii) 37 x 26
24 Find the products   
(i) (2a + 5b)(4a2 - 10ab + 25b2)
(ii) (2a - 5b)(4a2 + 10ab + 25b2)
25 Factorise
(i) 30x3y + 24x2y2 – 6xy
(ii) 5x(a – b) + 6y(a –b) 
26 Factorise
(i) 7a3 + 7a – 2a2 – 2
(ii) 4ax + 3by – 3ay –4bx
(iii) x3 + y3 + x2y + xy2
27 Factorise
(i) 9x2 + 30xy + 25y2
(ii) 9x2 – 30xy + 25y2
28 Factorize
(i) 9x2 – y2
(ii) (3 – x)2 – 36x2
(iii) (2x – 3y)2 – (3y + 4y)2
(iv) 16x4 – y4
29 Factorise
(i) x2 + y2 + 9z2 + 2xy + 6yz + 6xz,  
(ii) x2 + 4y2 + 9z2 – 4xy – 12yz + 6xz
30 Factorise x6 – y6