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Page No 99:

Question 1:

Factorise:
(i) 12x + 15
(ii) 14m − 21
(iii) 9n − 12n2

Answer:

 (i) 12x+15=34x+5
 (ii) 14m-21=72m-3
 (iii) 9n-12n2=3n3-4n

Page No 99:

Question 2:

Factorise:
(i) 16a2 − 24ab
(ii) 15ab2 − 20a2b
(iii) 12x2y3 − 21x3y2

Answer:

 (i) H.C.F. of 16a2 and 24ab is 8a.

        ∴ 16a2-24ab=8a2a-3b
     
(ii) H.C.F. of 15ab2 and 20a2b is 5ab.

        ∴ 15ab2-20a2b=5ab3b-4a

 (iii) ​H.C.F. of 12x2y3 and 21x3y2 is 3x2y2.

        ∴ 12x2y3-21x3y2=3x2y24y-7x

Page No 99:

Question 3:

Factorise:
(i) 24x3 − 36x2y
(ii) 10x3 − 15x2
(iii) 36x3y − 60x2y3z

Answer:

(i) H.C.F. of 24x3 and 36x2y is 6x2.

        ∴  24x3-36x2y=6x24x-6y

(ii)  H.C.F. of 10x3 and 15x2 is 5x3.

         ∴ 10x3-15x2=5x22x-3
   
(iii) H.C.F. of 36x3y and 60x2y3z is 12x2y.

         ∴ 36x3y-60x2y3z=12x2y3x-5y2z

Page No 99:

Question 4:

Factorise:
(i) 9x3 − 6x2 + 12x
(ii) 8x2 − 72xy + 12x
(iii) 18a3b3 − 27a2b3 + 36a3b2

Answer:

(i) H.C.F. of 9x36x2 and 12x is 3x.

        ∴ 9x3-6x2+12x=3x3x2-2x+4

(ii) H.C.F. of 8x372xy and 12x is 4x.

       ∴  8x3-72xy+12x=4x2x2-18y+3

(iii) H.C.F. of 18a3b327a2b3 and 36a3b2 is 9a2b2.

        ∴ 18a3b3-27a2b3+36a3b2=9a2b22ab-3b+4a

Page No 99:

Question 5:

Factorise:
(i) 14x3 + 21x4y − 28x2y2
(ii) −5 − 10t + 20t2

Answer:

(i) H.C.F. of 14x321x4y and 28x2y2 is 7x2.

        ∴ 14x3+21x4y-28x2y2=7x22x+3x2y-4y2

(ii) H.C.F. of -5-10t and 20t2 is 5.

         ∴ -5-10t+20t2=5-1-2t+4t2

Page No 99:

Question 6:

Factorise:
(i) x(x + 3) + 5(x + 3)
(ii) 5x(x − 4) − 7(x − 4)
(iii) 2m(1 − n) + 3(1 − n)

Answer:

(i) xx+3+5x+3=x+3x+5

(ii) 5xx-4-7x-4=x-45x-7

(iii) 2m1-n+31-n=1-n2m+3

Page No 99:

Question 7:

Factorise:
6a(a − 2b) + 5b(a − 2b)

Answer:

We have:
6aa-2b+5ba-2b=a-2b6a+5b

Page No 99:

Question 8:

Factorise:
x3(2ab) + x2(2ab)

Answer:

We have:
x32a-b+x22a-b=2a-bx3+x2=x2x+12a-b

Page No 99:

Question 9:

Factorise:
9a(3a − 5b) − 12a2(3a − 5b)

Answer:

We have:
9a3a-5b-12a23a-5b=3a-5b9a-12a2=3a3a-5b3-4a

Page No 99:

Question 10:

Factorise:
(x + 5)2 − 4(x + 5)

Answer:

We have:
x+52-4x+5=x+5x+5-4
                         =x+5x+5-4=x+5x+1

x+52-4x+5=x+5x+1
      

Page No 99:

Question 11:

Factorise:
3(a − 2b)2 − 5(a − 2b)

Answer:

 3a-2b2-5a-2b=a-2b3a-2b-5
                                    =a-2b3a-6b-5

∴ 3a-2b2-5a-2b=a-2b3a-6b-5

Page No 99:

Question 12:

Factorise:
2a + 6b − 3(a + 3b)2

Answer:

We have:
      2a+6b-3a+3b2=2a+3b-3a+3b2                             =a+3b2-3a+3b                             =a+3b2-3a-9b

2a+6b-3a+3b2=a+3b2-3a-9b

Page No 99:

Question 13:

Factorise:
16(2p − 3q)2 − 4(2p − 3q)

Answer:

 We have:
162p-3q2-42p-3q=2p-3q162p-3q-4                                      =2p-3q32p-48q-4

    ∴ 162p-3q2-42p-3q=2p-3q32p-48q-4

Page No 99:

Question 14:

Factorise:
x(a − 3) + y(3 − a)

Answer:

 We have:
xa-3+y3-a=xa-3-ya-3                          =a-3x-y

∴ xa-3+y3-a=a-3x-y                          

Page No 99:

Question 15:

Factorise:
12(2x − 3y)2 − 16(3y − 2x)

Answer:

We have:
122x-3y2-163y-2x=122x-3y2+162x-3y                                      =2x-3y122x-3y+16                                      =2x-3y24x-36y+16

 122x-3y2-163y-2x=2x-3y24x-36y+16                                                                            

Page No 99:

Question 16:

Factorise:
(x + y)(2x + 5) − (x + y)(x + 3)

Answer:

We have:
x+y2x+5-x+yx+3=x+y2x+5-x+3                                           =x+y2x+5-x-3                                           =x+yx+2 

Page No 99:

Question 17:

Factorise:
ar + br + at + bt

Answer:

By grouping the terms:
ar+br+at+bt=ar+br+at+bt                      =ra+b+ta+b                      =a+br+t

ar+br+at+bt=a+br+t                      

Page No 99:

Question 18:

Factorise:
x2axbx + ab

Answer:

 By suitably arranging the terms:
         x2-ax-bx+ab=x2-bx-ax+ab                        =x2-bx-ax-ab                        =xx-b-ax-b                        =x-bx-a

       ∴ x2-ax-bx+ab=x-bx-a                        

Page No 99:

Question 19:

Factorise:
ab2bc2ab + c2

Answer:

 By suitably arranging the terms:
       ab2-bc2-ab+c2=ab2-ab-bc2+c2                          =ab2-ab-bc2-c2                          =abb-1-c2b-1                          =b-1ab-c2

       ∴ ab2-bc2-ab+c2=b-1ab-c2                          

Page No 99:

Question 20:

Factorise:
x2xz + xyyz

Answer:

 By suitably arranging the terms:
      x2-xz+xy-yz=x2+xy-xz-yz                       =x2+xy-xz+yz                       =xx+y-zx+y                       =x+yx-z 

      ∴ x2-xz+xy-yz=x+yx-z                       

Page No 99:

Question 21:

Factorise:
6abb2 + 12ac − 2bc

Answer:

 By suitably arranging the terms:
      6ab-b2+12ac-2bc=6ab+12ac-b2-2bc                               =6ab+12ac-b2+2bc                               =6ab+2c-bb+2c                               =b+2c6a-b

     ∴ 6ab-b2+12ac-2bc=b+2c6a-b                               

Page No 99:

Question 22:

Factorise:
(x − 2y)2 + 4x − 8y

Answer:

 We have:
       x-2y2+4x-8y=x-2y2+4x-2y                          =x-2yx-2y+4x-2y                          =x-2yx-2y+4                          =x-2yx-2y+4

      ∴ x-2y2+4x-8y=x-2yx-2y+4                          

Page No 99:

Question 23:

Factorise:
y2xy(1 − x) − x3

Answer:

We have:
        y2-xy1-x-x3=y2-xy+x2y-x3                         =y2-xy+x2y-x3                         =yy-x+x2y-x                         =y-xy+x2

       ∴ y2-xy1-x-x3=y-xy+x2                         

Page No 99:

Question 24:

Factorise:
(ax + by)2 + (bxay)2

Answer:

We have:
        ax+by2+bx-ay2=a2x2+b2y2+2axby+b2x2+a2y2-2bxay                                =a2x2+a2y2+b2y2+b2x2+2axby-2bxay                                =a2x2+y2+b2x2+b2y2+2axby-2axby                                =a2x2+y2+b2x2+y2                                =x2+y2a2+b2

       ∴ ax+by2+bx-ay2=x2+y2a2+b2                                

Page No 99:

Question 25:

Factorise:
ab2 + (a − 1)b − 1

Answer:

We have:
       ab2+a-1b-1=ab2+ba-b-1                         =ab2+ba-b+1                         =abb+1-1b+1                         =b+1ab-1

      ∴ ab2+a-1b-1=b+1ab-1                         

Page No 99:

Question 26:

Factorise:
x3 − 3x2 + x − 3

Answer:

We have:
        x3-3x2+x-3=x3-3x2+x-3                     =x2x-3+1x-3                     =x-3x2+1

       ∴ x3-3x2+x-3=x-3x2+1                     

Page No 99:

Question 27:

Factorise:
ab(x2 + y2) − xy(a2 + b2)

Answer:

We have:
      abx2+y2-xya2+b2=abx2+aby2-a2xy-b2xy                                  =abx2-a2xy+aby2-b2xy                                  =axbx-ay+byay-bx                                  =axbx-ay-bybx-ay                                  =bx-ayax-by                           

      ∴ abx2+y2-xya2+b2=bx-ayax-by

Page No 99:

Question 28:

Factorise:
x2x(a + 2b) + 2ab

Answer:

We have:
        x2-xa+2b+2ab=x2-ax-2bx+2ab
                                       =x2-2bx-ax+2ab=x2-2bx-ax-2ab=xx-2b-ax-2b=x-2bx-a

      ∴ x2-xa+2b+2ab=x-2bx-a



Page No 100:

Question 1:

Factorise:
x2 − 36

Answer:

We have:
      x2-36=x2-62          =x+6x-6

      ∴ x2-36=x+6x-6          

Page No 100:

Question 2:

Factorise:
4a2 − 9

Answer:

 We have:
      4a2-9=2a2-32          =2a+32a-3

      ∴ 4a2-9=2a+32a-3          

Page No 100:

Question 3:

Factorise:
81 − 49x2

Answer:

We have:
       81-49x2=92-7x2             =9+7x9-7x

      ∴ 81-49x2=9+7x9-7x             

Page No 100:

Question 4:

Factorise:
4x2 − 9y2

Answer:

We have:
       4x2-9y2=2x2-3y2            =2x+3y2x-3y

      ∴ 4x2-9y2=2x+3y2x-3y            

Page No 100:

Question 5:

Factorise:
16a2 − 225b2

Answer:

We have:
       16a2-225b2=4a2-15b2                  =4a+15b4a-15b

      ∴ 16a2-225b2=4a+15b4a-15b                  

Page No 100:

Question 6:

Factorise:
9a2b2 − 25

Answer:

We have:
9a2b2-25=3ab2-52               =3ab+53ab-5       

     ∴ 9a2b2-25=3ab+53ab-5               

Page No 100:

Question 7:

Factorise:
16a2 − 144

Answer:

 We have:
         16a2-144=4a2-122               =4a+124a-12                = 4(a+ 3) 4(a-3) = 16 (a+3) (a-3)

       ∴ 16a2-144=16a+3a-3               

Page No 100:

Question 8:

Factorise:
63a2 − 112b2

Answer:

 We have:
        63a2-112b2=79a2-16b2                   =73a2-4b2                   =73a+4b3a-4b

       ∴ 63a2-112b2=73a+4b3a-4b                   

Page No 100:

Question 9:

Factorise:
20a2 − 45b2

Answer:

 We have:
       20a2-45b2=54a2-9b2                     =52a2-3b2                     =52a+3b2a-3b

      ∴ 20a2-45b2=52a+3b2a-3b                     

Page No 100:

Question 10:

Factorise:
12x2 − 27

Answer:

We have:
       12x2-27=34x2-9                 =32x2-32                 =32x+32x-3

      ∴ 12x2-27=32x+32x-3                 

Page No 100:

Question 11:

Factorise:
x3 − 64x

Answer:

We have:
       x3-64x=xx2-64              =xx2-82              =xx+8x-8

      ∴ x3-64x=xx+8x-8              

Page No 100:

Question 12:

Factorise:
16x5 − 144x3

Answer:

We have:
        16x5-144x3=16x3x2-9                       =16x3x2-32                       =16x3x+3x-3  

       ∴ 16x5-144x3=16x3x+3x-3                         

Page No 100:

Question 13:

Factorise:
3x5 − 48x3

Answer:

 We have:
      3x5-48x3=3x3x2-16                  =3x3x2-42                  =3x3x+4x-4

      ∴ 3x5-48x3=3x3x+4x-4                  

Page No 100:

Question 14:

Factorise:
16p3 − 4p

Answer:

We have:
      16p3-4p=4p4p2-1                 =4p2p2-12                 =4p2p+12p-1

      ∴ 16p3-4p=4p2p+12p-1                 

Page No 100:

Question 15:

Factorise:
63a2b2 − 7

Answer:

 We have:
      63a2b2-7=79a2b2-1                   =73ab2-12                   =73ab+13ab-1

     ∴ 63a2b2-7=73ab+13ab-1                   

Page No 100:

Question 16:

Factorise:
1 − (b − c)2

Answer:

 We have:
      1-b-c2=12-b-c2                   =1+b-c1-b-c                   =1+b-c1-b+c

      ∴ 1-b-c2=1+b-c1-b+c                   

Page No 100:

Question 17:

Factorise:
(2a + 3b)2 − 16c2

Answer:

We have:
     2a+3b2-16c2=2a+3b2-4c2                               =2a+3b+4c2a+3b-4c                               =2a+3b+4c2a+3b-4c

     ∴ 2a+3b2-16c2=2a+3b+4c2a+3b-4c                               

Page No 100:

Question 18:

Factorise:
(l + m)2 − (lm)2

Answer:

We have:
       l+m2-l-m2=l+m+l-ml+m-l-m                                =l+m+l-ml+m-l+m                                =2l2m

      ∴ l+m2-l-m2=2l2m                                

Page No 100:

Question 19:

Factorise:
(2x + 5y)2 − 1

Answer:

We have:
      2x+5y2-1=2x+5y2-12                        =2x+5y+12x+5y-1                        =2x+5y+12x+5y-1

      ∴ 2x+5y2-1=2x+5y+12x+5y-1                        

Page No 100:

Question 20:

Factorise:
36c2 − (5a + b)2

Answer:

 We have:
     36c2-5a+b2=6c2-5a+b2                            =6c+5a+b6c-5a+b                            =6c+5a+b6c-5a-b

      ∴ 36c2-5a+b2=6c+5a+b6c-5a-b                            

Page No 100:

Question 21:

Factorise:
(3x − 4y)2 − 25z2

Answer:

We have:
       3x-4y2-25z2=3x-4y2-5z2                          =3x-4y+5z3x-4y-5z                          =3x-4y+5z3x-4y-5z

      ∴ 3x-4y2-25z2=3x-4y+5z3x-4y-5z                        

Page No 100:

Question 22:

Factorise:
x2y2 − 2y − 1

Answer:

 We have:
      x2-y2-2y-1=x2-y2+2y+1   
                      =x2-y+12=x+y+1x-y+1=x+y+1x-y-1

      ∴ x2-y2-2y-1=x+y+1x-y-1   

Page No 100:

Question 23:

Factorise:
25 − a2b2 − 2ab

Answer:

We have:
      25-a2-b2-2ab=25-a2+b2+2ab
                                =25-a+b2=52-a+b2=5+a+b5-a+b=5+a+b5-a-b

      ∴ 25-a2-b2-2ab=5+a+b5-a-b

Page No 100:

Question 24:

Factorise:
25a2 − 4b2 + 28bc − 49c2

Answer:

 We have:
       25a2-4b2+28bc-49c2=25a2-4b2-28bc+49c2
                                           =5a2-2b-7c2=5a+2b-7c5a-2b-7c=5a+2b-7c5a-2b+7c

     ∴ 25a2-4b2+28bc-49c2=5a+2b-7c5a-2b+7c

Page No 100:

Question 25:

Factorise:
9a2b2 + 4b − 4

Answer:

We have:
      9a2-b2+4b-4=9a2-b2-4b+4
                             =3a2-b-22=3a+b-23a-b-2=3a+b-23a-b+2

     ∴ 9a2-b2+4b-4=3a+b-23a-b+2

Page No 100:

Question 26:

Factorise:
100 − (x − 5)2

Answer:

We have:
       100-x-52=102-x-52
                          =10+x-510-x-5=10+x-510-x+5=5+x15-x

      ∴ 100-x-52=5+x15-x

Page No 100:

Question 27:

Factorise:
Evaluate {(405)2 − (395)2}.

Answer:

 We have:
       4052-3952=405+395405-395                              =800×10                              =8000

      ∴ 4052-3952=8000                              

Page No 100:

Question 28:

Factorise:
Evaluate {(7.8)2 − (2.2)2}.

Answer:

We have:
      7.82-2.22=7.8+2.27.8-2.2                            =10×5.6                            =56

      ∴ 7.82-2.22=56



Page No 101:

Question 1:

Factorise:
x2 + 8x + 16

Answer:

We have:
     x2+8x+16=x2+2×x×4+42                    =x+42

     ∴ x2+8x+16=x+42                    

Page No 101:

Question 2:

Factorise:
x2 + 14x + 49

Answer:

We have:
      x2+14x+49=x2+2×x×7+72                      =x+72

      ∴ x2+14x+49=x+72                      

Page No 101:

Question 3:

Factorise:
1 + 2x + x2

Answer:

We have:
      1+2x+x2=x2+2x+1                  =x2+2×x×1+12                  =x+12

     ∴ 1+2x+x2=x+12                  

Page No 101:

Question 4:

Factorise:
9 + 6z + z2

Answer:

We have;
      9+6z+z2=z2+6z+9                  =z2+2×x×3+32                  =z+32

     ∴ 9+6z+z2=z+32                  

Page No 101:

Question 5:

Factorise:
x2 + 6ax + 9a2

Answer:

We have:
       x2+6ax+9a2=x2+2×x×3a+3a2                        =x+3a2

    ∴ x2+6ax+9a2=x+3a2                        

Page No 101:

Question 6:

Factorise:
4y2 + 20y + 25

Answer:

We have:
      4y2+20y+25=2y2+2×2y×5+52                         =2y+52

     ∴ 4y2+20y+25=2y+52                         

Page No 101:

Question 7:

Factorise:
36a2 + 36a + 9

Answer:

We have:
      36a2+36a+9=6a2+2×6a×3+32                      =6a+32

    ∴ 36a2+36a+9=6a+32                         

Page No 101:

Question 8:

Factorise:
9m2 + 24m + 16

Answer:

We have:
      9m2+24m+16=3m2+2×3m×4+42                        
                             =3m+42

     ∴ 9m2+24m+16=3m+42                  

Page No 101:

Question 9:

Factorise:
z2 + z + 14

Answer:

We have:
       z2+z+14=z2+2×z×12×122
                       =z+122

     ∴ z2+z+14=z+122

Page No 101:

Question 10:

Factorise:
49a2 + 84ab + 36b2

Answer:

We have:
       49a2+84ab+36b2=7a2+2×7a×6b+6b2
                                   =7a+6b2

     ∴ 49a2+84ab+36b2=7a+6b2

Page No 101:

Question 11:

Factorise:
p2 − 10p + 25

Answer:

We have:
       p2-10p+25=p2-2×p×5+52
                           =p-52

      ∴ p2-10p+25=p-52

Page No 101:

Question 12:

Factorise:
121a2 − 88ab + 16b2

Answer:

 We have:
       121a2-88ab+16b2=11a2-2×11a×4b+4b2
                                     =11a-4b2

     ∴ 121a2-88ab+16b2=11a-4b2

Page No 101:

Question 13:

Factorise:
1 − 6x + 9x2

Answer:

 We have:
      1-6x+9x2=9x2-6x+1
                       =3x2-2×3x×1+12=3x-12

      ∴ 1-6x+9x2=3x-12

Page No 101:

Question 14:

Factorise:
9y2 − 12y + 4

Answer:

 We have:
      9y2-12y+4=3y2-2×3y×2+22
                         =3y-22

    ∴ 9y2-12y+4=3y-22

Page No 101:

Question 15:

Factorise:
16x2 − 24x + 9

Answer:

We have:
       16x2-24x+9=4x2-2×4x×3+32
                           =4x-32

      ∴ 16x2-24x+9=4x-32

Page No 101:

Question 16:

Factorise:
m2 − 4mn + 4n2

Answer:

We have:
      m2-4mn+4n2=m2-2×m×2n+2n2
                            =m-2n2

     ∴ m2-4mn+4n2=m-2n2

Page No 101:

Question 17:

Factorise:
a2b2 − 6abc + 9c2

Answer:

We have:
      a2b2-6abc+9c2=ab2-2×ab×3c+3c2
                                =ab-3c2

Page No 101:

Question 18:

Factorise:
m4 + 2m2n2 + n4

Answer:

We have:
      m4+2m2n2+n4=m22+2×m2×n2+n22
                              =m2+n22

      ∴ m4+2m2n2+n4=m2+n22

Page No 101:

Question 19:

Factorise:
(l + m)2 − 4lm

Answer:

We have:
    l+m2-4lm=l2+m2+2lm-4lm
                         =l2+m2+2lm-4lm=l2+m2-2lm=l2+m2-2×l×m=l-m2

     ∴ l+m2-4lm=l-m2



Page No 102:

Question 1:

Factorise:
x2 + 5x + 6

Answer:

The given expression is x2+ 5x+ 6.Find two numbers that follow the conditions given below:Sum =5 Product =6Clearly, the numbers are 3 and 2. x2+ 5x+ 6 =  x2+ 3x + 2x + 6                       =x(x+3) +2 (x+3)                        =(x+3) (x+2)     

Page No 102:

Question 2:

Factorise:
y2 + 10y + 24

Answer:

The given expression is y2+ 10y + 24.Find two numbers that follow the conditions given below:Sum =10 Product =24Clearly, the numbers are 6 and 4.y2+ 10y + 24 =  y2+ 6y + 4y + 24                           = y (y+6) +4 (y+6)                           = (y+6) (y+4)     

Page No 102:

Question 3:

Factorise:
z2 + 12z + 27

Answer:

The given expression is z2+ 12z+ 27.Find two numbers that follow the conditions given below:Sum =12 Product =27Clearly, the numbers are 9 and 3. z2+ 12z+ 27 = z2+ 9z + 3z + 27                       =z(z+9) +3(z+9)                        =(z+9) (z+3)     

Page No 102:

Question 4:

Factorise:
p2 + 6p + 8

Answer:

The given expression is p2+ 6p + 8.Find two numbers that follow the conditions given below:Sum =6 Product =8Clearly, the numbers are 4 and 2. p2+ 6p + 8 =  p2+ 4p + 2p +8                       =p(p+4) +2 (p+4)                        =(p+4) (p+2)     

Page No 102:

Question 5:

Factorise:
x2 + 15x + 56

Answer:

The given expression is x2+ 15x + 56.Find two numbers that follow the conditions given below:Sum =15 Product =56Clearly, the numbers are 8 and 7. x2+ 15x + 56= x2+ 8x + 7x +56                          =x(x+8) +7 (x+8)                            =(x+8) (x+7)     

Page No 102:

Question 6:

Factorise:
y2 + 19y + 60

Answer:

The given expression is y2+ 19y + 60.Find two numbers that follow the conditions given below:Sum =19 Product =60Clearly, the numbers are 15 and 4.y2+ 19y + 60= y2+ 15y + 4y +60                          =y(y+15) +4 (y+15)                            =(y+15) (y+4)     

Page No 102:

Question 7:

Factorise:
x2 + 13x + 40

Answer:

The given expression is x2+ 13x + 40.Find two numbers that follow the conditions given below:Sum =13 Product =40Clearly, the numbers are 8 and 5.x2+ 13x + 40= x2+ 8x + 5x +40                          =x(x+8) +5 (x+8)                            =(x+8) (x+5)     

Page No 102:

Question 8:

Factorise:
q2 − 10q + 21

Answer:

The given expression is q2-10q + 21.Find two numbers that follow the conditions given below:Sum =-10 Product =21Clearly, the numbers are -7 and -3.q2-10q + 21= q2-7q -3q+ 21                          =q(q-7) -3 (q-7)                            =(q-7) (q-3)     

Page No 102:

Question 9:

Factorise:
p2 + 6p − 16

Answer:

The given expression is p2+ 6p -16.Find two numbers that follow the conditions given below:Sum =6 Product =-16Clearly, the numbers are 8 and -2. p2+ 6p -16=  p2+8p-2p -16                          =p(p+8) -2 (p+8)                            =(p+8) (p-2)     

Page No 102:

Question 10:

Factorise:
x2 − 10x + 24

Answer:

The given expression is x2-10x +24.Find two numbers that follow the conditions given below:Sum =-10 Product =24Clearly, the numbers are -6 and -4.x2-10x +24=  x2-6x-4x +24                        =x(x-6) -4 (x-6)                            =(x-6) (x-4)     

Page No 102:

Question 11:

Factorise:
x2 − 23x + 42

Answer:

The given expression is x2-23x + 42.Find two numbers that follow the conditions given below:Sum =-23 Product =42Clearly, the numbers are -21 and -2.x2-23x + 42=  x2-21x-2x + 42                        =x(x-21) -2 (x-21)                            =(x-21) (x-2)     

Page No 102:

Question 12:

Factorise:
x2 − 17x + 16

Answer:

The given expression is x2-17x + 16.Find two numbers that follow the conditions given below:Sum =-17 Product =16Clearly, the numbers are -16 and -1.x2-17x + 16=  x2-16x-x + 16                        =x(x-16) -1 (x-16)                            =(x-16) (x-1)     



Page No 103:

Question 13:

Factorise:
y2 − 21y + 90

Answer:

The given expression is y2-21y + 90.Find two numbers that follow the conditions given below:Sum =-21 Product =90Clearly, the numbers are -15 and -6.y2-21y + 90= y2-15y-6y + 90                        =y(y-15) -6 (y-15)                            =(y-15) (y-6)     

Page No 103:

Question 14:

Factorise:
x2 − 22x + 117

Answer:

The given expression is x2-22x + 117.Find two numbers that follow the conditions given below:Sum =-22 Product =117Clearly, the numbers are -13 and -9. x2-22x + 117=  x2-13x -9x+ 117                        =x(x-13) -9 (x-13)                            =(x-13) (x-9)     

Page No 103:

Question 15:

Factorise:
x2 − 9x + 20

Answer:

The given expression is x2-9x + 20.Find two numbers that follow the conditions given below:Sum =-9 Product =20Clearly, the numbers are -5 and -4. x2-9x + 20=   x2-5x-4x + 20                        =x(x-5) -4 (x-5)                            =(x-5) (x-4)     

Page No 103:

Question 16:

Factorise:
x2 + x − 132

Answer:

The given expression is x2+x -132.Find two numbers that follow the conditions given below:Sum =1 and pProduct =-132Clearly, the numbers are 12 and -11.x2+ x -132=  x2+12x-11x -132                        =x(x+12) -11 (x+12)                            =(x+12) (x-11)     

Page No 103:

Question 17:

Factorise:
x2 + 5x − 104

Answer:

The given expression is x2+5x -104.Find two numbers that follow the conditions given below:Sum =5 Product =-104Clearly, the numbers are 13 and -8.x2+5x -104=  x2+13x-8x -104                        =x(x+13) -8 (x+13)                            =(x+13) (x-8)     

Page No 103:

Question 18:

Factorise:
y2 + 7y − 144

Answer:

The given expression is y2+7y -144.Find two numbers that follow the conditions given below:Sum =7 Product =-144Clearly, the numbers are 16 and -9.y2+7y -144= y2+16y-9y -144                        =y(y+16) -9 (y+16)                            =(y+16) (y-9)     

Page No 103:

Question 19:

Factorise:
z2 + 19z − 150

Answer:

The given expression is z2+19z -150.Find two numbers that follow the conditions given below:Sum =19 Product =-150Clearly, the numbers are 25 and -6.z2+19z -150= z2+25z-6z -150                        =z(z+25) -6 (z+25)                            =(z+25) (z-6)     

Page No 103:

Question 20:

Factorise:
y2 + y − 72

Answer:

The given expression is y2+y -72.Find two numbers that follow the conditions given below:Sum =1 Product =-72Clearly, the numbers are 9 and -8. y2+y -72=  y2+9y-8y -72                        =y(y+9) -8 (y+9)                            =(y+9) (y-8)     

Page No 103:

Question 21:

Factorise:
a2 + 6a − 91

Answer:

The given expression is a2+6a -91.Find two numbers that follow the conditions given below:Sum =6 Product =-91Clearly, the numbers are 13 and -7.a2+6a -91=  a2+13a-7a -91                        =a(a+13) -7 (a+13)                            =(a+13) (a-7)     

Page No 103:

Question 22:

Factorise:
p2 − 4p − 77

Answer:

The given expression is p2-4p -77.Find two numbers that follow the conditions given below:Sum =-4 Product =-77Clearly, the numbers are -11 and 7.p2-4p -77=  p2-11p+7p -77                        =p(p-11) +7 (p-11)                         =(p-11) (p+7)     

Page No 103:

Question 23:

Factorise:
x2 − 7x − 30

Answer:

The given expression is x2-7x -30.Find two numbers that follow the conditions given below:Sum =-7 Product =-30Clearly, the numbers are -10 and 3. x2-7x -30=   x2-10x+3x -30                        =x(x-10) +3 (x-10)                         =(x-10) (x+3)     

Page No 103:

Question 24:

Factorise:
x2 − 11x − 42

Answer:

The given expression is x2-11x -42.Find two numbers that follow the conditions given below:Sum =-11 Product =-42Clearly, the numbers are -14 and 3. x2-11x -42=   x2-14x+3x -42                        =x(x-14) +3 (x-14)                         =(x-14) (x+3)     

Page No 103:

Question 25:

Factorise:
x2 − 5x − 24

Answer:

The given expression is x2-5x -24.Find two numbers that follow the conditions given below:Sum =-5 Product =-24Clearly, the numbers are -8 and 3.x2-5x -24=   x2-8x+3x -24                        =x(x-8) +3 (x-8)                         =(x-8) (x+3)     

Page No 103:

Question 26:

Factorise:
y2 − 6y − 135

Answer:

The given expression is y2-6y -135.Find two numbers that follow the conditions given below:Sum =-6 Product =-135Clearly, the numbers are -15 and 9.y2-6y -135=   y2-15y+9y -135                        =y(y-15) +9 (y-15)                         =(y-15) (y+9)     

Page No 103:

Question 27:

Factorise:
z2 − 12z − 45

Answer:

The given expression is z2-12z -45.Find two numbers that follow the conditions given below:Sum =-12 Product =-45Clearly, the numbers are -15 and 3. z2-12z -45=    z2-15z+3z -45                        =z(z-15) +3 (z-15)                         =(z-15) (z+3)     

Page No 103:

Question 28:

Factorise:
x2 − 4x − 12

Answer:

The given expression is x2-4x -12.Find two numbers that follow the conditions given below:Sum =-4 Product =-12Clearly, the numbers are -6 and 2.x2-4x -12=   x2-6x +2x -12                        =x(x-6) +2 (x-6)                         =(x-6) (x+2)     

Page No 103:

Question 29:

Factorise:
3x2 + 10x + 8

Answer:

The given expression is 3x2+10x +8.Find two numbers that follow the conditions given below:Sum =10Product =3×8=24Clearly, the numbers are 6 and 4.3x2+10x +8=  3x2+10x +8                        =3x2+6x+4x +8                         =3x(x+2)+4 (x+2)                           =(x+2) (3x +4 )       

Page No 103:

Question 30:

Factorise:
3y2 + 14y + 8

Answer:

The given expression is 3y2+14y +8Find two numbers that follow the conditions given below:Sum =14 Product=24Clearly, the numbers are 12 and 2.3y2+14y +8= 3y2+12y+2y +8                         =3y(y+4)+2 (y+4)                           =(3y+2) (y+4 )       

Page No 103:

Question 31:

Factorise:
3z2 − 10z + 8

Answer:

The given expression is 3z2-10z +8.Find two numbers that follow the conditions given below:Sum =-10 Product =3×8=24Clearly, the numbers are -6 and -4.3z2-10z +8=  3z2-6z-4z +8                        = 3z(z-2)-4(z -2)                         =(3z-4)(z-2)                              

Page No 103:

Question 32:

Factorise:
2x2 + x − 45

Answer:

The given expression is 2x2+x -45.Find two numbers that follow the conditions given below:Sum =1 Product =-45×2=-90Clearly, the numbers are 10 and -9.2x2+x -45=  2x2+10x-9x -45                      = 2x(x +5) -9(x+5)                      =(2x-9)(x+5)                              

Page No 103:

Question 33:

Factorise:
6p2 + 11p − 10

Answer:

The given expression is 6p2+11p -10.Find two numbers that follow the conditions given below:Sum =11 Product =6×-10=-60Clearly, the numbers are 15 and -4.6p2+11p -10=  6p2+15p -4p-10                          = 3p(2p +5) -2(2p+5)                          =(2p+5)(3p-2)                              

Page No 103:

Question 34:

Factorise:
2x2 − 17x − 30

Answer:

The given expression is 2x2-17x -30.Find two numbers that follow the conditions given below:Sum =-17 Product =-30×2=-60Clearly, the numbers are -20 and 3.2x2-17x -30=  2x2-20x+3x -30                         = 2x(x -10) +3(x-10)                         =(2x+3)(x-10)                              

Page No 103:

Question 35:

Factorise:
7y2 − 19y − 6

Answer:

The given expression is 7y2-19y -6.Find two numbers that follow the conditions given below:Sum =-19 Product =7×-6=-42Clearly, the numbers are -21 and 2.7y2-19y -6= 7y2-21y+2y -6                         = 7y(y -3)+2(y-3)                         =(7y+2)(y-3)                              

Page No 103:

Question 36:

Factorise:
28 − 31x − 5x2

Answer:

The given expression is 28-31x-5x2.Find two numbers that follow the conditions given below:Sum =-31Product =28×-5=-140Clearly, the numbers are -35 and 4.28-31x-5x2= 28+4x-35x-5x2                         = 4(x+7)-5x(7+x)                         =(x+7) (4-5x)                              

Page No 103:

Question 37:

Factorise:
3 + 23z − 8z2

Answer:

The given expression is 3+23z-8z2.Find two numbers that follow the conditions given below:Sum = 23  Product = 3×-8=-24Clearly, the numbers are 24 and -1.3+23z-8z2= 3+24z-z-8z2                         = 3(1+8z)-z(1+8z)                         =(1+8z) (3-z)                              

Page No 103:

Question 38:

Factorise:
6x2 − 5x − 6

Answer:

The given expression is 6x2-5x -6.Find two numbers that follow the conditions given below:Sum =-5 Product =-6×6=-36Clearly, the numbers are -9 and 4.6x2-5x -6= 6x2-9x+4x -6                         = 3x(2x-3)+2(2x-3)                         =(2x-3) (3x+2)                              

Page No 103:

Question 39:

Factorise:
3m2 + 24m + 36

Answer:

The given expression is 3m2+24m +36.Find two numbers that follow the conditions given below:Sum =24 Product =36×3=108Clearly, the numbers are 18 and 6.3m2+24m +36= 3m2+18m+6m +36                             = 3m(m+6)+6(m+6)                              =(3m+6) (m+6)=3m+2m+6                              

Page No 103:

Question 40:

Factorise:
4n2 − 8n + 3

Answer:

The given expression is 4n2-8n +3.Find two numbers that follow the conditions given below:Sum =-8 Product =4×3=12Clearly, the numbers are -6 and -2.4n2-8n +3=4n2-2n -6n+3                      = 2n(2n-1)-3(2n-1)                      =(2n-1) (2n-3)                              

Page No 103:

Question 41:

Factorise:
6x2 − 17x − 3

Answer:

The given expression is 6x2-17x-3.Find two numbers that follow the conditions given below:Sum =-17 Product = 6×-3=-18Clearly, the numbers are -18 and 1.6x2-17x-3=6x2-18x+x-3                      = 6x(x-3)+1(x-3)                      =(6x+1) (x-3)                              

Page No 103:

Question 42:

Factorise:
7x2 − 19x − 6

Answer:

The given expression is 7x2-19x-6.Find two numbers that follow the conditions given below: Sum =-19 Product =7×-6=-42Clearly, the numbers are -21 and 2.7x2-19x-6=7x2-21x+2x-6                      = 7x(x-3)+2(x-3)                      =(7x+2) (x-3)                              

Page No 103:

Question 1:

Tick (✓) the correct answer:
(7a2 − 63b2) = ?
(a) (7a − 9b)(9a + 7b)
(b) (7a − 9b)(7a + 9b)
(c) 9(a − 3b)(a + 3b)
(d) 7(a − 3b)(a + 3b)

Answer:

(d) 7(a − 3b)(a + 3b)

(7a2 -63b2)=7(a2 -9b2)                       =7(a -3b)(a+3b)

Page No 103:

Question 2:

Tick (✓) the correct answer:
(2x − 32x3) = ?
(a) 2(x − 4)(x + 4)
(b) 2x(1 − 2x)2
(c) 2x(1 + 2x)2
(d) 2x(1 − 4x)(1 + 4x)

Answer:

(d) 2x(1 − 4x)(1 + 4x)

(2x-32x3)=2x(1-16x2)=2x(1-4x)(1+4x)

Page No 103:

Question 3:

Tick (✓) the correct answer:
x3 − 144x = ?
(a) x(x − 12)2
(b) x(x + 12)2
(c) x(x − 12)(x + 12)
(d) none of these

Answer:

(c) x(x − 12)(x + 12)

x3 - 144x=x(x2 -144)=x(x-12)(x+12)

Page No 103:

Question 4:

Tick (✓) the correct answer:
(2 − 50x2) = ?
(a) 2(1 − 5x)2
(b) 2(1 + 5x)2
(c) (2 − 5x)(2 + 5x)
(d) 2(1 − 5x)(1 + 5x)

Answer:

(d) 2(1 − 5x)(1 + 5x)

2-50x2 =2(1-25x2)=2(1-5x)(1+5x)

Page No 103:

Question 5:

Tick (✓) the correct answer:
a2 + bc + ab + ac = ?
(a) (a + b)(a + c)
(b) (a + b)(b + c)
(c) (b + c)(c + a)
(d) a(a + b + c)

Answer:

(a) (a + b)(a + c)

a2+bc +ab +ac=a2+ab+bc +ac=a(a+b)+c(a+b)=(a+c)(a+b)

Page No 103:

Question 6:

Tick (✓) the correct answer:
pq2 + q(p − 1) − 1 = ?
(a) (pq + 1)(q − 1)
(b) p(q + 1)(q − 1)
(c) q(p − 1)(q + 1)
(d) (pq − 1) (q + 1)

Answer:

(d) (pq − 1) (q + 1)

pq2+q(p-1)-1=pq2+qp-q-1=pq(q+1)-1(q+1)=(pq-1)(q+1)

Page No 103:

Question 7:

Tick (✓) the correct answer:
abmn + anbm = ?
(a) (ab)(mn)
(b) (am)(b + n)
(c) (an)(m + b)
(d) (ma)(nb)

Answer:

(b) (am)(b + n)

ab-mn+an-bm=ab+an-mn-bm=a(b+n)-m(n+b)=(a-m)(b+n)

Page No 103:

Question 8:

Tick (✓) the correct answer:
abab + 1 = ?
(a) (a − 1)(b − 1)
(b) (1 − a)(1 − b)
(c) (a − 1)(1 − b)
(d) (1 − a)(b − 1)

Answer:

(a) (a − 1)(b − 1)

ab-a-b+1=a(b-1)-1(b-1)=(a-1)(b-1)

Page No 103:

Question 9:

Tick (✓) the correct answer:
x2xz + xyyz = ?
(a) (xy)(x + z)
(b) (xy)(x z)
(c) (x + y)(xz)
(d) (xy)(z − x)

Answer:

(c) (x + y)(xz)

x2-xz+xy-yz=x(x-z)+y(x-z)=(x+y) (x-z)



Page No 104:

Question 10:

Tick (✓) the correct answer:
(12m2 − 27) = ?
(a) (2m − 3)(3m − 9)
(b) 3(2m − 9)(3m − 1)
(c) 3(2m − 3)(2m + 3)
(d) none of these

Answer:

(c) 3(2m − 3)(2m + 3)

12m2-27=3(4m2-9)=3 (2m-3)(2m+3)

Page No 104:

Question 11:

Tick (✓) the correct answer:
x3x = ?
(a) x(x2x)
(b) x(xx2)
(c) x(1 + x)(1 − x)
(d) x(x + 1)(x − 1)

Answer:

(d) x(x + 1)(x − 1)

x3-x=x( x2-1)=x(x-1)(x+1)

Page No 104:

Question 12:

Tick (✓) the correct answer:
1 − 2ab − (a2 + b2) = ?
(a) (1 + ab)(1 + a + b)
(b) (1 + a + b)(1 − a + b)
(c) (1 + a + b)(1 − ab)
(d) (1 + a b)(1 − a + b)

Answer:

(c) (1 + a + b)(1 − ab)

1-2ab-(a2 +b2)=1-2ab -a2-b2=1-(2ab +a2+b2)=1-(a+b)2=(1-a-b)(1+a+b)

Page No 104:

Question 13:

Tick (✓) the correct answer:
x2 + 6x + 8 = ?
(a) (x + 3)(x + 5)
(b) (x + 3)(x + 4)
(c) (x + 2)(x + 4)
(d) (x + 1)(x + 8)

Answer:

(c) (x + 2)(x + 4)

x2+6x+8=x2+4x+2x+8=x(x+4)+2(x+4)=(x+2)(x+4)

Page No 104:

Question 14:

Tick (✓) the correct answer:
x2 + 4x − 21 = ?
(a) (x − 7)(x + 3)
(b) (x + 7)(x − 3)
(c) (x − 7)(x − 3)
(d) (x + 7)(x + 3)

Answer:

(b) (x + 7)(x − 3)

x2+4x-21=x2+7x-3x-21=x(x+7)-3(x+7)=(x-3)(x+7)

Page No 104:

Question 15:

Tick (✓) the correct answer:
y2 + 2y − 3 = ?
(a) (y − 1)(y + 3)
(b) (y + 1)(y − 3)
(c) (y − 1)(y − 3)
(d) (y + 2)(y − 3)

Answer:

(a) (y − 1)(y + 3)

y2+2y-3=y2+3y-y-3=y(y+3)-1(y+3)=(y-1)(y+3)

Page No 104:

Question 16:

Tick (✓) the correct answer:
40 + 3xx2 = ?
(a) (x + 5)(x − 8)
(b) (5 − x)(8 + x)
(c) (5 + x)(8 − x)
(d) (5 − x)(8 − x)

Answer:

(c) (5 + x)(8 − x)

40+3x-x2=40+8x-5x-x2=8(5+x)-x(5+x)=(8-x)(x+5)

Page No 104:

Question 17:

Tick (✓) the correct answer:
2x2 + 5x + 3 = ?
(a) (x + 3)(2x + 1)
(b) (x + 1)(2x + 3)
(c) (2x + 5)(x − 3)
(d) none of these

Answer:

(b) (x + 1)(2x + 3)

2x2+5x+3=2x2+2x+3x+3=2x(x+1) +3(x+1)=(2x +3) (x+1)

Page No 104:

Question 18:

Tick (✓) the correct answer:
6a2 − 13a + 6 = ?
(a) (2a + 3)(3a − 2)
(b) (2a − 3)(3a + 2)
(c) (3a − 2)(2a − 3)
(d) (3a + 1)(2a − 3)

Answer:

(c) (3a − 2)(2a − 3)

6a2-13a+6=6a2-9a-4a+6=3a(2a-3)-2(2a-3)=(3a-2)(2a-3)

Page No 104:

Question 19:

Tick (✓) the correct answer:
4z2 − 8z + 3 = ?
(a) (2z − 1)(2z − 3)
(b) (2z + 1)(3 − 2z)
(c) (2z + 3)(3z − 1)
(c) (z − 1)(4z − 3)

Answer:

(a) (2z − 1)(2z − 3)

4z2-8z+3=4z2-6z-2z+3=2z(2z-3)-1(2z-3)=(2z-1)(2z-3)

Page No 104:

Question 20:

Tick (✓) the correct answer:
3 + 23y − 8y2 = ?
(a) (1 − 8y)(3 + y)
(b) (1 + 8y)(3 − y)
(c) (1 − 8y)(y − 3)
(d) (8y − 1)(y + 3)

Answer:

(b) (1 + 8y)(3 − y)

3+23y-8y2=3+24y-y-8y2=3(1+8y)-y(1+8y)=(3-y)(1+8y)



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