CLASS 8 MATHS 14 Factorisation QUESTIONS

 14 Factorisation

14.1 Introduction 

 14. 1.1 Factors Of Natural Numbers

14.1.2 Factors Of Algebraic Expressions

14.2 What Is Factorization

14.2.1 Method Of Common Factors

14.2.2 Factorization By Regrouping Terms

14.2.3 Factorization Using Identities

14.2 4  Factors Of The Form (X + A)( X + B)

14.3 Division Of Algebraic Expressions

14.3.1 Division Of Monomial By Another Monomial

14.3 2 Division Of A Polynomial By A Monomial

14.4 Division Of Algebraic Expressions Continued Polynomial By Polynomial

14.5 Can You Find The Error

WHAT HAVE WE DISCUSSED? 

1. When we factorise an expression, we write it as a product of factors. These factors may be numbers, algebraic variables or algebraic expressions. 

2. An irreducible factor is a factor which cannot be expressed further as a product of factors. 3. A systematic way of factorising an expression is the common factor method. It consists of three steps: (i) Write each term of the expression as a product of irreducible factors (ii) Look for and separate the common factors and (iii) Combine the remaining factors in each term in accordance with the distributive law. 

4. Sometimes, all the terms in a given expression do not have a common factor; but the terms can be grouped in such a way that all the terms in each group have a common factor. When we do this, there emerges a common factor across all the groups leading to the required factorisation of the expression. This is the method of regrouping. 

5. In factorisation by regrouping, we should remember that any regrouping (i.e., rearrangement) of the terms in the given expression may not lead to factorisation. We must observe the expression and come out with the desired regrouping by trial and error. 

6. A number of expressions to be factorised are of the form or can be put into the form : a2 + 2 ab + b2, a2 – 2ab + b2, a2 – b2 and x2 + (a + b) + ab. These expressions can be easily factorised using Identities I, II, III and IV, given in Chapter 9, 

a2 + 2 ab + b2 = (a + b)2 

a2 – 2ab + b2 = (a – b)2 

a2 – b2 = (a + b) (a – b) 

x2 + (a + b) x + ab = (x + a) (x + b) 

7. In expressions which have factors of the type (x + a) (x + b), remember the numerical term gives ab. Its factors, a and b, should be so chosen that their sum, with signs taken care of, is the coefficient of x.

8. We know that in the case of numbers, division is the inverse of multiplication. This idea is applicable also to the division of algebraic expressions.

9. In the case of division of a polynomial by a monomial, we may carry out the division either by dividing each term of the polynomial by the monomial or by the common factor method. 10. In the case of division of a polynomial by a polynomial, we cannot proceed by dividing each term in the dividend polynomial by the divisor polynomial. Instead, we factorise both the polynomials and cancel their common factors. 

11. In the case of divisions of algebraic expressions that we studied in this chapter, we have Dividend = Divisor × Quotient. 

In general, however, the relation is 

Dividend = Divisor × Quotient + Remainder 

Example 1: Factorise 12a2b + 15ab2

Example 2: Factorise 10x2 – 18x3 + 14x4

Factorise: (i) 12x + 36 (ii) 22y – 33z (iii) 14pq+35pqr

Example 3: Factorise 6xy – 4y + 6 – 9x. 

Example 4: Factorise x2 + 8x + 16

Example 5: Factorise 4y2 – 12y + 9

Example 6: Factorise 49p2 – 36 

Example 7: Factorise a2 – 2ab + b2 – c2 

Example 8: Factorise m4 – 256

Example 9: Factorise x2 + 5x + 6

Example 10: Find the factors of y2 –7y +12. 

Example 11: Obtain the factors of z2 – 4z – 12. 

Example 12: Find the factors of 3m2 + 9m + 6. 

Example 13: Do the following divisions. 

(i) –20x4 ⎟ 10x2(ii) 7x2y2z2 ⎟ 14xyz 

Divide. (i) 24xy2z3 by 6yz2(ii) 63a2b4c6 by 7a2b2c3

Example 14: Divide 24(x2yz + xy2z + xyz2) by 8xyz

Example 15: Divide 44(x4 – 5x3 – 24x2) by 11x (x – 8) 

Example 16: Divide z(5z2 – 80) by 5z(z + 4)

EXERCISE 14.1 

1. Find the common factors of the given terms. 

(i) 12x, 36

(ii) 2y, 22xy

(iii) 14 pq, 28p2q2 

(iv) 2x, 3x2, 4

(v) 6 abc, 24ab2, 12 a2b 

(vi) 16 x3, – 4x2, 32x

(vii) 10 pq, 20qr, 30rp 

(viii) 3x2 y3, 10x3 y2,6 x2 y2z 

2. Factorise the following expressions. 

(i) 7x – 42

(ii) 6p – 12q

(iii) 7a2 + 14a 

(iv) – 16 z + 20 z3

(v) 20 l2 m + 30 a l m 

(vi) 5 x2 y – 15 xy2

(vii) 10 a2 – 15 b2 + 20 c2 

(viii) – 4 a2 + 4 ab – 4 ca

(ix) x2 y z + x y2z + x y z2 

(x) a x2 y + b x y2 + c x y z 

3. Factorise.

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

(ii) 15 xy – 6x + 5y – 2 

(iii) ax + bx – ay – by

(iv) 15 pq + 15 + 9q + 25p 

(v) z – 7 + 7 x y – x y z

EXERCISE 14.2 

1. Factorise the following expressions. 

(i) a2 + 8a + 16

(ii) p2 – 10 p + 25

(iii) 25m2 + 30m + 9 

(iv) 49y2 + 84yz + 36z2

(v) 4x2 – 8x + 4 

(vi) 121b2 – 88bc + 16c2 

(vii) (l + m)2 – 4lm (Hint: Expand ( l + m)2first) 

(viii) a4 + 2a2b2 + b4 

2. Factorise. 

(i) 4p2 – 9q2

(ii) 63a2 – 112b2

(iii) 49x2 – 36 

(iv) 16x5 – 144x3

(v) (l + m)2 – (l – m)2 

(vi) 9x2 y2 – 16

(vii) (x2 – 2xy + y2) – z2 

(viii) 25a2 – 4b2 + 28bc – 49c2

3. Factorise the expressions. 

(i) ax2 + bx

(ii) 7p2 + 21q2

(iii) 2x3 + 2xy2 + 2xz2 

(iv) am2 + bm2 + bn2 + an2

(v) (lm + l) + m + 1 

4. Factorise. 

(i) a4 – b4

(ii) p4 – 81

(iii) x4 – (y + z)4 

(iv) x4 – (x – z)4

(v) a4 – 2a2b2 + b4 

5. Factorise the following expressions. 

(i) p2 + 6p + 8

(ii) q2 – 10q + 21

(iii) p2 + 6p – 16

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

(vii) 5y2 – 20y – 8z + 2yz 

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

(ix) 6xy – 4y + 6 – 9x

EXERCISE 14.3 

1. Carry out the following divisions. 

(i) 28x4 ➗ 56x

(ii) –36y3 ➗ 9y2

(iii) 66pq2r3 ➗ 11qr2 

(iv) 34x3y3z3 ➗51xy2z3

(v) 12a8b8 ➗ (– 6a6b4) 

2. Divide the given polynomial by the given monomial. 

(i) (5x2 – 6x) ➗3x

(ii) (3y8 – 4y6 + 5y4) ➗ y4 

(iii) 8(x3y2z2 + x2y3z2 + x2y2z3) ➗ 4x2y2z2

(iv) (x3 + 2x2 +3x) ➗ 2x 

(v) (p3q6 – p6q3) ➗ p3q3 

3. Work out the following divisions.

(i) (10x – 25) ➗ 5

(ii) (10x – 25) ➗ (2x – 5) 

(iii) 10y(6y + 21) ➗ 5(2y + 7)

(iv) 9x2y2(3z – 24) ➗ 27xy(z – 8)

(v) 96abc(3a – 12) (5b – 30) ➗ 144(a – 4) (b – 6) 

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)

5. Factorise 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)

EXERCISE 14.4 

Find and correct the errors in the following mathematical statements.

1. 4(x – 5) = 4x – 5

2. x(3x + 2) = 3x2 + 2

3. 2x + 3y = 5xy

4. x + 2x + 3x = 5x

5. 5y + 2y + y – 7y = 0

6. 3x + 2x = 5x2

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

8. (2x)2 + 5x = 4x + 5x = 9x

9. (3x + 2)2 = 3x2 + 6x + 4

10. 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 = – 2 

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

11. (y – 3)2 = y2 – 9

12. (z + 5)2 = z2 + 25 

13. (2a + 3b) (a – b) = 2a2 – 3b2

14. (a + 4) (a + 2) = a2 + 8  

15. (a – 4) (a – 2) = a2 – 8

16. 3x² / 3x² =0

Class 8 Slip Test -3 for Algebraic Expressions and Factorization 50 marks

 

Slip Test -3 CLICK below Download button TO DOWNLOAD THE QUESTION PAPER Class- 8                                                                                                                         Subject - Maths  Chapters - Algebraic Expressions and Factorization                                                    Marks : 20 +30 =50 marks Section A  Short Answers type1 of 2 marks each  Add: 4y (3y2 + 5y – 7) and 2 (y3 – 4y2 + 5) Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively: (10m,5n) Simplify 3x (4x – 5) + 3 and find its values for (i) x = 3 (ii) x = 1/2. Factorize   p2-10p+25  Divide as directed.  5(2x + 1) (3x + 5) ÷ (2x + 1)  Factorise:  4p² – 9q² Multiply  (a² + b) (a + b² ) Factorise:15 pq + 15 + 9q + 25p Obtain the volume of rectangular boxes with the following length, breadth and height respectively.  5a, 3a² , 7a4 Find the product.  (5 – 2x) (3 + x) Section B  Short Answers type2 of 3 marks each Simplify (a + b) (2a – 3b + c) – (2a – 3b) c. Subtract 3l(l-4m+5n) from 4l(10n-3m+2l) Simplify (a + b) (2a – 3b + c) – (2a – 3b) c Simplify: (a + b) (c – d) + (a – b) (c + d) + 2 (ac + bd)  Simplify:(1.5x – 4y)(1.5x + 4y + 3) – 4.5x + 12y  Simplify: (x + y)(x2 – xy + y2)  Simplify: (a + b + c)(a + b – c) Factorise the expressions and divide them as directed.  (y ² + 7y + 10) ÷ (y + 5) Factorise the expressions. am² + bm² + bn² + an² Factorise: a4 – 2a²b²  + b4

Class 8 Slip Test -2 for Exponents and Powers and Direct and Inverse Proportions 20 marks

 
Slip Test -2
Class- 8                                            Subject - Maths                                     Marks : 20  
Chapters -  Exponents and Powers and Direct and Inverse Proportions                                      
Section A  MCQs of 1 mark each
The value of (-2/3)4 is equal to:
(a) 16/81                  (b) 81/16             (c) -16/81            (d) 81/ −16
am x an = ______
          a) am+n             b) am-n         c) amn           d) am/n
 The  value of 1x0 is
        a) 0 b) 1 c) 1x d-1x
Express the number 3.02 x 10-6 in usual form.
       (a) 0.00000302   (b) 0.30200000   (c) 0.0000000302         (d) 302000000
The approximate distance of the moon from the earth is 384,467,000 m and in  exponential form. This distance can be written as 
               (a) 3.84,467 × 108m (b) 384,467 × 10-8 m (c) 384,467 × 10-9 m (d) 3.844,67 × 10-13 m 
If 8 men can do a piece of work in 20 days, in how many days will 20 men complete the same work?
       a) 8                       b)  10                c) 12                          d) 20   
If  x and y vary inversely from each other, and x= 20 when y=3. Find y when x = 12.
         (a) y=10                (b) y=5    (c) y=0                 (d) y=6 
If Anandi types 200 words in half an hour, she will be able to type ______________ words in 12 minutes. (Assuming the speed of typing to be uniform.) 
          a) 40                   b) 60             c) 80                d) 100
The scale of a map is given as 1: 300. Two cities are 4 km apart on the map. The actual distance between them is:
a) 1000 km    b) 1100 km     c) 1200 km      d) 1300 km
If x and y are inversely proportional then find the value of a.
a) 4                       b)  1                             c)  3           d) 2 
Section B  Very Short Answer of 1 mark each
A machine in a soft drink factory fills 840 bottles in six hours. How many bottles will it  fill in two hours? 
 Express Charge of an electron is 0.000,000,000,000,000,000,16 coulomb in standard  form. 
The scientific notation of 1035.3 is ____
The multiplicative inverse of 3-3 is_______.
The product   of  y² x 2y 22  x 5y26  is ___
 The value of   is ___
Express 4−3 as a power with base 2. 
A loaded truck travels 168 km in 5 hours. How far can it travel in 25 minutes?
 If ‘x’ and ‘y’ are in a direct proportion then  x/y=constant is correct. True / False
What is the value of (−1)−1? 

Answer key 

Section A  MCQs of 1 mark each

1. (a) 16/81                  

2.   a) am+n         

3. b) 1

4.  (a) 0.00000302   

5. (a) 3.84,467 × 108m 

6.  a) 8                      

7. (b) y=5  

8. c) 80           

9. c) 1200 km      

10. d) 2 

Section B  Very Short Answer of 1 mark each

11. 280

12.  1.6 x 10 -19

13. 1.0353 x 10 3

14. 1/27

15. 10y50

16. 1/220

17. 2-6

18. 14 km

19. True

20. (-1)