Class 8 - Maths - Factorisation

Exercise 14.1

Question 1:

Find the common factors of the given terms.

(i) 12x, 36          (ii) 2y, 22xy         (iii) 14pq, 28 p2 q2      (iv) 2x, 3x2, 4       (v) 6abc, 24ab2, 12a2

(vi) 16x3, -4x3, 32x            (vii) 10pq, 20qr, 30rp             (viii) 3x2 y3, 10x3 y3, 6 x2 y2 z

Answer:

(i) 12x = 2 * 2 * 3 * x  

     36 = 2 * 2 * 3 * 3

Hence, the common factors are 2, 2 and 3 = 2 * 2 * 3 = 12      

(ii) 2y = 2 * y

     22xy = 2 * 11 * x * y

Hence, the common factors are 2 and y = 2 * y = 2y    

(iii) 14pq * 2 * 7 * p * q

      28 p2 q2 = 2 * 2 * 7 * p * p * q * q

Hence, the common factors are 2 * 7 * p * q = 14pq 

(iv) 2x =2 * x * 1

      3x2 = 3 * x * x * 1

      4 = 2 * 2 * 1

Hence, the common factors is 1   

(v) 6abc = 2 * 2 * a * b * c

     24ab2 = 2 * 2 * 3 * a * a *b

    12a2 b = 2 * 2 * 3 * a * a * b

Hence, the common factors are 2 * 3 * a * b = 6ab

(vi) 16x3 = 2 * 2 * 2 * 2 * x * x * x

      -4x3 = (-1) * 2 * 2 * x * x * x

      32x = 2 * 2 * 2 * 2 * 2 * x

Hence, the common factors are 2 * 2 * x = 4x

(vii) 10pq = 2 * 5 * p * q

        20qr = 2 * 2 * 5 * q * r

       30rp = 2 * 3 * 5 * r * p

Hence, the common factors are 2 * 5 = 10

(viii) 3x2 y3 = 3 * x * x * y * y * y  

        10x3 y3 = 2 * 5 * x * x * x * y * y * y

        6 x2 y2 z = 2 * 3 * x * x * y * y * z

Hence, the common factors are x * x * y * y = x2 y2

Question 2:

Factorize the following expressions.

(i) 7x – 42        (ii) 6p – 12q       (iii) 7a2 + 14a           (iv) -16z + 20z3              (v) 20l2 m + 30alm

(vi) 5x2 y – 15xy2         (vii) 10a2 – 15b2 + 20c2         (viii) -4a2 + 4ab – 4ca    (ix) x2 yz + xy2 z + xyz2

(x) ax2 y + bxy2 + cxyz

Answer:

(i) 7x – 42 = 7 * x – 7 * 6 = 7(x - 6)

(ii) 6p – 12q = 2 * 3 * p – 2 * 2 * 3 * q

                      = 2 * 3(p – 2 * q)

                      = 6(p -2q)

(iii) 7a2 + 14a = 7 * a * a + 2 * 7 * a

                        = 7 * a(a + 2)

                      = 7a(a + 2)

(iv) -16z + 20z3 = (-1) * 2 * 2 * 2 * 2 * z + 2 * 2 * 5 * z * z * z

                            = 2 * 2 * z(-2 * 2 + 5 * z * z)

                           = 4z(-4 + 5z2)

(v) 20l2 m + 30alm = 2 * 2 * 5 * l * l * m + 2 * 3 * 5 * a * l * m

                                  = 2 * 5 * l *m(2 * l + 3 * a)

                                  = 10lm(2l + 3a)

(vi) 5x2 y – 15xy2 = 5 * x * x * y – 3 * 5 * x * y * y

                              = 5 * x * y(x – 3 * y)

                              = 5xy(x – 3y)

(vii) 10a2 – 15b2 + 20c2 = 2 * 5 * a * a – 3 * 5 * b * b + 2 * 2 * 5 * c * c

                                         = 5(2 * a * a – 3 * b * b + 2 * 2 * c * c)

                                         = 5(2a2 – 3b2 + 4c2)

(viii) -4a2 + 4ab – 4ca = (-1) * 2 * 2 * a * a + 2 * 2 * a * b – 2 * 2 * c * a

                                      = 2 *2 * a (-a + b - c)

                                      = 4a(-a + b - c)

(ix) x2 yz + xy2 z + xyz2 = x * x * y * z + x * y * y * x + x * y * z * z

                                        = x * y * z(x + y + z)

                                        = xyz(x + y + z)

(x) ax2 y + bxy2 + cxyz = a * x * x * y + b * x * y * y + c * x * y * z

                                       = x * y(a * x + b * y + c * z)

                                       = xy(ax + by + cz)

Question 3:

Factorize:

(i) x2 + xy + 8x + 8y                          (ii) 15xy – 6x + 5y – 2               (iii) ax + bx – ay – by

(iv) 15pq + 15 + 9q + 25p               (v) z – 7 + 7xy – xyz

Answer:

(i) x2 + xy + 8x + 8y = x(x + y) + 8(x + y)

                                  = (x + y)(x + 8)                          

(ii) 15xy – 6x + 5y – 2 = 3x(5y - 2) + 1(5y - 2)

                                       = (5y - 2)(3x + 1)             

(iii) ax + bx – ay – by = x(a + b) – y(a + b)

                                      = (a + b)(x - y)  

(iv) 15pq + 15 + 9q + 25p = 15pq + 25p + 15 + 9q  

                                             = 5p(3q + 5) + 3(3q + 5)

                                             = (3q + 5)(5p + 3)          

(v) z – 7 + 7xy – xyz = 7xy – 7 – xyz + z

                                   = 7(xy - 1) – z(xy - 1)

                                   = (xy - 1)(7 - z)      

 

                                                  Exercise 14.2

Question 1:

Factorize the following expressions:

(i) a2 + 8a + 16         (ii) p2 – 10p + 25            (iii) 25m2 + 30m + 9                (iv) 49y2 + 84yz + 36z2

(v) 4x2 – 8x + 4        (vi) 121b2 – 88bc + 16c2      (vii) (l + m)2 – 4lm             (viii) a4 + 2a2 b2 + b4

Answer:

(i) a2 + 8a + 16 = a2 + (4 + 4)a + 4 * 4

                          = (a + 4)(a + 4)                            [x2 + (a + b) + ab = (x + a)(x + b)]

                         = (a + 4)2       

(ii) p2 – 10p + 25 = p2 – (-5 - 5)p + (-5)(-5)

                              = (p - 5)(p - 5)

                              = (p - 5)2                                   [x2 + (a + b) + ab = (x + a)(x + b)]

(iii) 25m2 + 30m + 9 = (5m)2 + 2 * 5m * 3 + 32

                                    = (5m + 3)2                        [a2 + 2ab + b2 = (a + b)2]

(iv) 49y2 + 84yz + 36z2 = (7y)2 + 2 * 7y * 6z + (6z)2

                                        = (7y + 6z)2                    [a2 + 2ab + b2 = (a + b)2]

(v) 4x2 – 8x + 4 = (2x)2 – 2 * 2x * 2 + 22

                           = (2x - 2)2                                   [a2 - 2ab + b2 = (a - b)2]

                           = 22 (x - 1)2

                           = 4(x - 1)2   

(vi) 121b2 – 88bc + 16c2 = (11b)2 – 2 * 11b * 4c + (4c)2    

                                           = (11b – 4c)2             [a2 - 2ab + b2 = (a - b)2]

(vii) (l + m)2 – 4lm = l2 + 2 * l * m + m2 – 4lm

                               = l2 + 2lm + m2 – 4lm

                               = l2 - 2lm + m2  

                               = (l - m)2                               [a2 - 2ab + b2 = (a - b)2]

(viii) a4 + 2a2 b2 + b4 = (a2)2 + 2a2 b2 + (b2)2

                                    = (a2 + b2)2                         [a2 + 2ab + b2 = (a + b)2]

Question 2:

Factorize:

(i) 4p2 – 9q2         (ii) 63a2 – 112b2         (iii) 49x2 – 36      (iv)  16x5 – 144x2       (v) (l + m)2 – (l - m)2

(vi) 9x2 y2 – 16           (vii) (x2 – 2xy + y2) – z2             (viii) 25a2 – 4b2 + 28bc – 49c2

Answer:

(i) 4p2 – 9q2 = (2p)2 – (3q)2

                      = (2p – 3q)(2p + 3q)                [a2 – b2 = (a - b)(a + b)] 

(ii) 63a2 – 112b2 = 7(9a2 – 16b2 )

                             = 7{(3a)2 – (4b)2}

                             = 7(3a – 4b)(3a + 4b)       [a2 – b2 = (a - b)(a + b)]

(iii) 49x2 – 36 = (7x)2 – 62 

                        = (7x - 6)(7x + 6)                     [a2 – b2 = (a - b)(a + b)]

(iv)  16x5 – 144x2 = 16x3 (x2 – 9)

                              = 16x3 (x2 – 32

                              = 16x3 (x – 3)(x + 3)         [a2 – b2 = (a - b)(a + b)]

(v) (l + m)2 – (l - m)2 = l2 + 2lm + m2 – (l2 – 2lm + m2)

                                    = l2 + 2lm + m2 – l2 + 2lm - m2

                                    = 4lm    

(vi) 9x2 y2 – 16 = (3xy)2 – 42

                          = (3xy - 4)(3xy + 4)                          [a2 – b2 = (a - b)(a + b)]

(vii) (x2 – 2xy + y2) – z2 = (x - y)2 – z2

                                        = (x – y + z)(x – y - z)         [a2 – b2 = (a - b)(a + b)]  

(viii) 25a2 – 4b2 + 28bc – 49c2 = 25a2 – (4b2 - 28bc + 49c2)

                                                    = 25a2 – {(2b)2 – 2 *2b * 7c + (7c)2}

                                                    = (5a)2 – (2b – 7c)2

                                                    = {5a - (2b – 7c)}(5a + 2b – 7c)

                                                    = {5a - 2b + 7c)(5a + 2b – 7c)

Question 3:

Factorize the expressions:

(i) ax2 + bx             (ii)  7p2 + 21q2            (iii)  2x3 + 2xy2 + 2xz2                (iv) am2 + bm2 + bn2 + an

(v)  (lm + l) + m + 1    (vi) y(y + z) + 9(y + z)      (vii) 5y2 – 20y – 8z + 2yz   (viii) 10ab + 4a + 5b + 2

(ix) 6xy – 4y + 6 – 9x

Answer:

(i) ax2 + bx = x(ax + b)            

(ii)  7p2 + 21q2 = 7(p2 + 3q2 )         

(iii)  2x3 + 2xy2 + 2xz2 = 2x(x2 + y2 + z2)              

(iv) am2 + bm2 + bn2 + an = m2(a + b) + n2(a + b)

                                             = (a + b)(m2 + n2)

(v)  (lm + l) + m + 1 = l(m + 1) + 1(m + 1)  

                                  = (m + 1)(l + 1)  

(vi) y(y + z) + 9(y + z) = (y + z)(y + 9)    

(vii) 5y2 – 20y – 8z + 2yz = 5y2 – 20y + 2yz – 8z

                                           = 5y(y - 4) + 2z(y - 4)

                                           = (y - 4)(5y + 2z)               

(viii) 10ab + 4a + 5b + 2 = 2a(5b + 2) + 1(5b + 2)

                                          = (5b + 2)(2a + 1)

(ix) 6xy – 4y + 6 – 9x = 6xy – 4y + 6 – 9x

                                     = 3x(2y – 3) – 2(2y - 3)

                                     = (2y - 3)(3x - 2)    

Question 4:

Factorize:

(i) a4 – b4         (ii) p4 – 81         (iii) x4 – (y + z)4          (iv) x4 – (x - z)4          (v) a4 – 2a2 b2 + b4

Answer:

(i) a4 – b4 = (a2)2 – (b2)2 = (a2 – b2)(a2 + b2)                    [a2 – b2 = (a - b)(a + b)]

                                         = (a - b)(a + b)(a2 + b2)             [a2 – b2 = (a - b)(a + b)]

(ii) p4 – 81 = (p2) – 92

                   = (p2 - 9)(p2 + 9)                                              [a2 – b2 = (a - b)(a + b)]

                   = (p2 - 32)(p2 + 9)        

                   = (p - 3)(p + 3)(p2 + 9)                                    [a2 – b2 = (a - b)(a + b)]

(iii) x4 – (y + z)4 = (x2)2 - [(y + z)2]2        

                            = [x2 – (y + z)2] [x2 + (y + z)2]

                            = [x – (y + z)][x + (y + z)] [x2 + (y + z)2]   [a2 – b2 = (a - b)(a + b)]

                            = [x – y - z)][x + y + z] [x2 + (y + z)2]        [a2 – b2 = (a - b)(a + b)] 

(iv) x4 – (x - z)4 = (x2)2 – [(x - z)2]2        

                        = [x2 – (x – z)2] [x2 + (x – z)2]                                    [a2 – b2 = (a - b)(a + b)] 

                        = [x – (x – z)] [x + (x – z)] [x2 + (x – z)2]

                        = [x – x + z] [x + x – z] [x2 + x2 – 2 * x * z + z2]    [(a - b)2 = a2 – 2ab + b2]

                        = z(2x - z)(2x2 – 2xz + z2)

(v) a4 – 2a2 b2 + b4 = (a2)2 – 2a2 b2 + (b2)2                [(a - b)2 = a2 – 2ab + b2]

                                 = (a2 – b2)2

                                 = [(a - b)(a + b)]2                        [a2 – b2 = (a - b)(a + b)] 

                                 = (a - b)2(a + b)2

Question 5:

Factorize the following expressions:

(i) p2 + 6p + 8                     (ii) q2 – 10q + 21                       (iii) p2 + 6p – 16

Answer:

(i) p2 + 6p + 8 = p2 + (4 + 2)p + 4 * 2

                         = p2 + 4p + 2p + 4 * 2

                         = p(p + 4) + 2(p + 4)

                         = (p + 4)(p + 2)                      

(ii) q2 – 10q + 21 = q2 – (7 + 3)q + 7 * 3

                              = q2 – 7q – 3q + 7 * 3

                              = q(q - 7) – 3(q - 7)

                              = (q - 7)(q - 3)             

(iii) p2 + 6p – 16 = p2 + (8 - 2)p – 8 * 2

                             = p2 + 8p – 2p – 8 * 2

                             = p(p + 8) – 2(p + 8) = (p + 8)(p - 2)    

                                                                  Exercise 14.3

Question 1:

Carry out the following divisions:

(i) 28x4 ÷ 56x                                (ii) -36y3 ÷ 9y2                             (iii) 66pq2 r3 ÷ 11qr2 

(iv) 34x3 y3 z3 ÷  51xy2 z3            (v) 12a8 b8 ÷ (-6a6 b4)

Answer:

(i) 28x4 ÷ 56x = 28x4 / 56x = (28/56) * (x4/x) = (1/2) * x3 = x3/2                               

(ii) -36y3 ÷ 9y2 = -36y3 / 9y2 = (-36/9) * (y3/y2) = -4y                          

(iii) 66pq2 r3 ÷ 11qr2 = 66pq2 r3 / 11qr2 = (66/11) * (pq2 r3 / qr2) = 6pqr

(iv) 34x3 y3 z3 ÷  51xy2 z3 = 34x3 y3 z3 /  51xy2 z3

                                           = (34/51) * (x3 y3 z3 /xy2 z3)    

                                           = (2/3) x2 y

                                           = 2x2 y/3     

(v) 12a8 b8 ÷ (-6a6 b4) = 12a8 b8 / (-6a6 b4)

                                      = (12/-6) * (a8 b8 /a6 b4)

                                      = -2a2 b4

Question 2:

Divide the given polynomial by the given monomial:

(i) (5x2 – 6x) ÷ 3x       (ii) (3y8 – 4y6 + 5y4) ÷  y4       (iii) 8(x3 y2 z2 + x2 y3 z2 + x2 y2 3z) ÷4 x2 y2 z2

(iv) (x3 + 2x2 + 3x) ÷  2x      (v) (p3 q6 – p6 q3) ÷ p3 q3

Answer:

(i) (5x2 – 6x) ÷ 3x = (5x2 – 6x)/3x

                              = 5x2/3x – 6x/3x

                           = 5x/3 – 2

                           = (5x - 6)/3      

(ii) (3y8 – 4y6 + 5y4) ÷ y4 = (3y8 – 4y6 + 5y4)/y4     

                                          = 3y8/y4 – 4y6/y4 + 5y4/ y4                     

                                          = 3y4 – 4y2 + 5

(iii) 8(x3 y2 z2 + x2 y3 z2 + x2 y2 3z) ÷ 4 x2 y2 z2

       = 8(x3 y2 z2 + x2 y3 z2 + x2 y2 3z)/4 x2 y2 z2           

       = 8x3 y2 z2/4 x2 y2 z2 + 8x2 y3 z2/4 x2 y2 z2 + 8x2 y2 3z/4 x2 y2 z2

       = 2x + 2y + 2z

       = 2(x + y + z)      

(iv) (x3 + 2x2 + 3x) ÷ 2x = (x3 + 2x2 + 3x)/2x    

                                        = x3/2x + 2x2/2x + 3x/2x

                                        = x2/2 + x + 3/2

                                        = (x2 + 2x + 3)/2

(v) (p3 q6 – p6 q3) ÷ p3 q3 = (p3 q6 – p6 q3)/p3 q3

                                           = p3 q6/p3 q3 – p6 q3/p3 q3

                                           = q3 – p3

Question 3:

Work out the following divisions:

(i) (10x - 25) ÷ 5                        (ii) (10x - 25) ÷ (2x - 5)               (iii) 10y(6y + 21) ÷  5(2y + 7)

(iv) 9x2 y2(3z - 24) ÷ 27xy(z - 8)        (v) 96abc(3a - 12)(5b - 30) ÷  144(a - 4)(a - 6)

Answer:

(i) (10x - 25) ÷ 5 = (10x - 25)/5

                          = 10x/5 – 25/5

                          = 2x - 5                       

(ii) (10x - 25) ÷ (2x - 5) = (10x - 25)/(2x - 5)

                                        = 5(2x - 5)/(2x - 5)

                                        = 5              

(iii) 10y(6y + 21) ÷  5(2y + 7) = 10y(6y + 21)/{5(2y + 7)}

                                                   = 2y(6y + 21)/(2y + 7)

                                                   = {2y * 3(2y + 7)}/(2y + 7)

                                                   = 2y * 3

                                                   = 6y

(iv) 9x2 y2(3z - 24) ÷ 27xy(z - 8) = 9x2 y2(3z - 24)/27xy(z - 8)      

                                                       = (9/27){xy * xy * 3(z - 8)}/{xy(z - 8)}

                                                       = xy     

(v) 96abc(3a - 12)(5b - 30) ÷ 144(a - 4)(a - 6) = 96abc(3a - 12)(5b - 30)/144(a - 4)(a - 6)

                                                                               = {96abc * 3(a - 4) * 5(b - 6)}/{144(a - 4)(a - 6)}

                                                                               = (96abc * 3 * 4)/144

                                                                               = 10abc

Question 4:

Divide as directed:

(i) 5(2x + 1)(3x + 5) ÷ (2x + 1)                                               (ii) 26xy(x + 5)(y - 4) ÷ 13x(y - 4)

(iii) 52pqr(p + q)(q + r)(r + p) ÷ 104pq(q + r)(r + p)          (iv) 20(y + 4)(y2 + 5y + 3) ÷ 5(y + 4)

(v) x(x + 1)(x + 2)(x + 3) ÷ x(x + 1)

 

Answer:

(i) 5(2x + 1)(3x + 5) ÷ (2x + 1) = {5(2x + 1)(3x + 5)}/(2x + 1)

                                                    = 5(3x + 5)     

(ii) 26xy(x + 5)(y - 4) ÷ 13x(y - 4) = {26xy(x + 5)(y - 4)}/{13x(y - 4)}

                                                         = 2y(x + 5)

(iii)   52pqr(p + q)(q + r)(r + p) ÷ 104pq(q + r)(r + p)

     = {52pqr(p + q)(q + r)(r + p)}/{104pq(q + r)(r + p)}

     = {52pqr(p + q)(q + r)(r + p)}/{52 * 2pq(q + r)(r + p)}

     = r(r + p)/2

(iv) 20(y + 4)(y2 + 5y + 3) ÷ 5(y + 4) = {20(y + 4)(y2 + 5y + 3)}/{5(y + 4)}

                                                              = 4(y2 + 5y + 3)

(v) x(x + 1)(x + 2)(x + 3) ÷ x(x + 1) = {x(x + 1)(x + 2)(x + 3)}/{x(x + 1)}

                                                           = (x + 2)(x + 3)

Question 5:

Factorize the expressions and divide them as directed:

(i) (y2 + 7y + 10) ÷ (y + 5)

(ii) (m2 – 14m - 32) ÷ (m + 2)

(iii) (5p2 – 25p + 20) ÷ (p - 1)

(iv) 4yz(z2 + 6z - 16) ÷ 2y(z + 8)

(v) 5pq(p2 – q2) ÷ 2p(p + q)

(vi) 12xy(9x2 – 16y2) ÷ 4xy(3x + 4y)

(vii) 39y3(50y2 - 98) ÷ 26y2(5y + 7)

 

Answer:

(i) (y2 + 7y + 10) ÷ (y + 5) = (y2 + 7y + 10)/(y + 5)

                                            = {y2 + (2 + 5)y + 2 * 5}/(y + 5)

                                            = {y2 + 2y + 5y + 2 * 5}/(y + 5)

                                            = {y(y + 2) + 5(y + 2)}/(y + 5)

                                            = {(y + 2)(y + 5)}/(y + 5)

                                            = y + 2

(ii) (m2 – 14m - 32) ÷ (m + 2) = (m2 – 14m - 32)/(m + 2)

                                                   = {m2 + (-16 + 2)m – 16 * 2)}/(m + 2)

                                                   = {m2 – 16m + 2m – 16 * 2)}/(m + 2)} 

                                                   = {m(m - 16) + 2(m – 16)}/(m + 2)

                                                   = {(m - 16)(m + 2)}/(m + 2)

                                                   = m + 2

(iii) (5p2 – 25p + 20) ÷ (p - 1) = (5p2 – 25p + 20)/(p - 1)

                                                   = 5(p2 – 5p + 4)/(p - 1)

                                                   = 5{p2 – (1 + 4)p + 4}/(p - 1)

                                                   = 5{p2 – p - 4p + 4}/(p - 1)

                                                   = 5{p(p - 1) – 4(p - 1)}/(p - 1)

                                                   = 5{(p - 1)(p - 4)}/(p - 1)

                                                   = 5(p - 4)     

(iv) 4yz(z2 + 6z - 16) ÷ 2y(z + 8) = {4yz(z2 + 6z - 16)}/{2y(z + 8)}

                                                       = {4yz(z2 + (8 - 2)z – 8 * 2)}/{2y(z + 8)}

                                                       = {4yz(z2 + 8z - 2z – 8 * 2)}/{2y(z + 8)}

                                                     = [4yz{z(z + 8) – 2(z + 8)}]/{2y(z + 8)}

                                                     = [4yz(z + 8)(z - 2)]/ {2y(z + 8)}

                                                     = 2z(z - 2)             

(v) 5pq(p2 – q2) ÷ 2p(p + q) = {5pq(p2 – q2)}/{2p(p + q)}

                                                = {5pq(p – q)(p + q)}/{2p(p + q)}

                                                = 5q(p – q)/2          

(vi) 12xy(9x2 – 16y2) ÷ 4xy(3x + 4y) = 12xy{(3x)2 – (4y)2}/{4xy(3x + 4y)}

                                                               = 12xy{(3x – 4y)(3x + 4y)}/{4xy(3x + 4y)}

                                                               = 3(3x – 4y)    

(vii) 39y3(50y2 - 98) ÷ 26y2(5y + 7) = {39y3(50y2 - 98)}/{26y2(5y + 7)}

                                                             = {39y3 * 2(25y2 - 49)}/{26y2(5y + 7)}

                                                             = [39y3 * 2{(5y)2 – 72}]/{26y2(5y + 7)}

                                                             = [39y3 * 2{(5y - 7)(5y + 7)}]/{26y2(5y + 7)}

                                                             = [13 * 3y3 * 2{(5y - 7)(5y + 7)}]/{13 * 2y2(5y + 7)}    

                                                             = 3y(5y - 7)

  

 

                                                                      Exercise 14.4

Question 1:

Find and correct the errors in the following mathematical statement: 4(x – 5) = 4x - 5

Answer:

L.H.S. = 4(x – 5) = 4x – 4 * 5 = 4x – 20

RHS: 4x – 5

Since LHS ≠ RHS

Hence, the correct mathematical statements is 4(x - 5) = 4x – 20

Question 2:

Find and correct the errors in the following mathematical statement: x(3x + 2) = 3x2 + 2

Answer:

L.H.S. = x(3x + 2) = 3x2 + 2x

R.H.S. = 3x2 + 2

Since LHS ≠ RHS

Hence, the correct mathematical statements is x(3x + 2) = 3x2 + 2x

Question 3:

Find and correct the errors in the following mathematical statement: 2x + 3y = 5xy

Answer:

L.H.S. = 2x + 3y

RHS = 5xy

Since LHS ≠ RHS

Hence, the correct mathematical statements is 2x + 3y = 5xy

 

 

Question 4:

Find and correct the errors in the following mathematical statement: x + 2x + 3x = 5x

Answer:

L.H.S. = x + 2x + 3x = x(1 + 2 + 3) = 6x

R.H.S. = 5x

Since LHS ≠ RHS

Hence, the correct mathematical statements is x + 2x + 3x = 6x

Question 5:

Find and correct the errors in the following mathematical statement: 5y + 2y + y – 7y = 0

Answer:

L.H.S. = 5y + 2y + y – 7y = 8y – 7y = y

R.H.S. = 0

Since LHS ≠ RHS

Hence, the correct mathematical statements is 5y + 2y + y – 7y = y

Question 6:

Find and correct the errors in the following mathematical statement: 3x + 2x = 5x2

Answer:

L.H.S. = 5x + 2x = 5x

R.H.S. = 5x2

Since LHS ≠ RHS

Hence, the correct mathematical statements is 3x + 2x = 5x

Question 7:

Find and correct the errors in the following mathematical statement:

(2x)2 + 4(2x) + 7 = 2x2 + 8x + 7

Answer:

L.H.S. = (2x)2 + 4(2x) + 7 = 4x2 + 8x + 7

R.H.S. = 2x2 + 8x + 7

Since LHS ≠ RHS

Hence, the correct mathematical statements is (2x)2 + 4(2x) + 7 = 2x2 + 8x + 7

Question 8:

Find and correct the errors in the following mathematical statement:

(2x)2 + 5x = 4x + 5x = 9x

Answer 8:

L.H.S. = (2x)2 + 5x = 4x2 + 5x

R.H.S. = 9x

Since LHS ≠ RHS

Hence, the correct mathematical statements is (2x)2 + 5x = 4x2 + 5x

Question 9:

Find and correct the errors in the following mathematical statement:

(3x + 2)2 = 3x2 + 6x + 9

Answer:

L.H.S. = (3x + 2)2 = (3x)2 + 2 * 3x * 2 + 22 = 9x2 + 12x + 4

R.H.S. = 3x2 + 6x + 9

Since LHS ≠ RHS

Hence, the correct mathematical statements is (3x + 2)2 = 9x2 + 12x + 4

 

Question 10:

Find and correct the errors in the following mathematical statements:

Substituting x = -3 in:

(a) x2 + 5x + 4 gives (-3)2 + 5(-3) + 4 = 9 + 2 + 4 = 15

(b) x2 - 5x + 4 gives (-3)2 - 5(-3) + 4 = 9 - 15 + 4 = 15

(c) x2 + 5x + 4 gives (-3)2 + 5(-3) = -9 - 15 = -24

Answer:

(a) Given expression is : x2 + 5x + 4

LHS: Put x = -3, we get

(-3)2 + 5(-3) + 4 = 9 - 15 + 4 = 13 – 15 = -2 ≠ RHS

Hence, x2 + 5x + 4 gives (-3)2 + 5(-3) + 4 = 9 - 15 + 4 = -2

 

(b) Given expression is: x2 - 5x + 4

LHS:

Put x = -3, we get

(-3)2 - 5(-3) + 4 = 9 + 15 + 4 = 28 ≠ RHS

Hence, x2 - 5x + 4 gives (-3)2 + 5(-3) + 4 = 9 + 15 + 4 = 28

 

(c) Given expression is: x2 + 5x 

LHS:

Put x = -3, we get

(-3)2 + 5(-3) = 9 - 15 = -6 ≠ RHS

Hence, x2 + 5x gives (-3)2 + 5(-3) = 9 - 15 = -6

Question 11:

Find and correct the errors in the following mathematical statement: (y - 3)2 = y2 – 9

Answer:

LHS: (y - 3)2 = y2 – 2 * y * 3 + 32

                      = y2 – 6y + 9 ≠ RHS

Hence, the correct statement is: (y - 3)2 = y2 – 6y + 9

Question 12:

Find and correct the errors in the following mathematical statement: (z + 5)2 = z2 + 25

Answer:

LHS: (z + 5)2 = z2 + 2 * z * 5 + 52

                      = z2 + 10z + 25 ≠ RHS

Hence, the correct statement is: (z + 5)2 = z2 + 10z + 25

Question 13:

Find and correct the errors in the following mathematical statement:

(2a + 3b)(a - b) = 2a2 – 3b2

Answer:

LHS: (2a + 3b)(a - b) = 2a(a - b) + 3b(a - b)

                                    = 2a2 – 2ab + 3ab – 3b2

                                    = 2a2 + ab – 3b2 ≠ RHS

Hence, the correct statement is: (2a + 3b)(a - b) = 2a2 + ab – 3b2

 

Question 14:

Find and correct the errors in the following mathematical statement:

(a + 4)(a + 2) = a2 + 8

Answer:

LHS: (a + 4)(a + 2) = a(a + 2) + 4(a + 2)

                                = a2 + 2a + 4a + 8

                                = a2 + 6a + 8 ≠ RHS

Hence, the correct statement is: (a + 4)(a + 2) = a2 + 6a + 8

Question 15:

Find and correct the errors in the following mathematical statement:

(a - 4)(a - 2) = a2 - 8

Answer:

LHS: (a - 4)(a - 2) = a(a - 2) - 4(a - 2)

                                = a2 - 2a - 4a + 8

                                = a2 - 6a + 8 ≠ RHS

Hence, the correct statement is: (a - 4)(a - 2) = a2 - 6a + 8

Question 16:

Find and correct the errors in the following mathematical statement: 3x2/3x2 = 0

Answer:

L.H.S. = 3x2/3x2 = 1/1 = 1 ≠ RHS

Hence, the correct statement is: 3x2/3x2 = 1

 

Question 17:

Find and correct the errors in the following mathematical statement: (3x2 + 1)/3x2 = 1 + 1 = 2

Answer:

L.H.S. = (3x2 + 1)/3x2 = 3x2/3x2 + 1/3x2 = 1 + 1/3x2 ≠ RHS

Hence, the correct statement is: (3x2 + 1)/3x2 = 1 + 1/3x2

Question 18:

Find and correct the errors in the following mathematical statement: 3x/(3x + 2) = 1/2

Answer:

L.H.S. = 3x/(3x + 2) ≠ RHS

Hence, the correct statement is: 3x/(3x + 2) = 3x/(3x + 2)

Question 19:

Find and correct the errors in the following mathematical statement: 3/(4x + 3) = 1/4x

Answer:

L.H.S. = 3/(4x + 3) ≠ RHS

Hence, the correct statement is: 3/(4x + 3) = 3/(4x + 3)

Question 20:

Find and correct the errors in the following mathematical statement: (4x + 5)/4x = 5

Answer:

L.H.S. = (4x + 5)/4x = 4x/4x + 5/4x = 1 + 5/4x ≠ RHS

Hence, the correct statement is: (4x + 5)/4x = 1 + 5/4x

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