1.Find the Factors of Each of the Following Numbers

Chapter 3More about Factorization 3.1

Chapter 3More about Factorization

Warm-up Exercise

1.Find the factors of each of the following numbers.

(a)12(b)72

2.Expand the following expressions.

(a)(x7)2(b)(3x4y)2

3.Expand the following expressions.

(a)(2a5)2(b)(6a2b)2

4.Expand the following expressions.

(a)(x4y)(x4y)(b)(9x7y)(9x7y)

5.Expand the following expressions.

(a)(x2)(3x4)(b)(3x2y)(4x3y)

6.Express the following expressions in positive indices.

(a)(b)

Build-up Exercise

[ This part provides two extra sets of questions for each exercise in the textbook, namely Elementary Setand Advanced Set. You may choose to complete any ONE set according to your need. ]

Exercise 3A

Elementary Set

Level 1

1.Factorize the following expressions.

(a)2a4(b)3a3

(c)55a(d)18a9

(e)4a26(f)7025a

2.Factorize the following expressions.

(a)4a8(b)3a9

(c)48b(d)8a2b

(e)3ab12(f)166ab

3.Factorize the following expressions.

(a)4x12y8(b)9x27y18z

(c)5x10y20z(d)3x18y27z

(e)16x4y20z(f)6x10y22z

4.Factorize the following expressions.

(a)5a10ab(b)4a8ab

(c)12ab20a(d)18ab9bc

(e)28ab49b2(f)50ab60b2

5.Factorize the following expressions.

(a)4a2b8ab2(b)25ab2100a2b

(c)4a310a2b(d)2ab4a2b2

(e)3a2b9b2c2(f)10a2bc220ab2c

6.Factorize the following expressions.

(a)b(a3)2(a3)(b)a(a1)4b(a1)

(c)15(ab)b(ba)(d)8(a2b)x(a2b)y

(e)(a12)25b(12a)(f)2b(a2b)(a2b)2

7.Factorize the following expressions.

(a)xyaxay(b)2b2aba

(c)xyx2y2(d)2xax2yby

(e)xy2xz2yz(f)3ax3ayxy

8.Factorize the following expressions.

(a)p29(b)100q2

(c)p2a2b2(d)16p21

(e)5q245(f)3p212

Level 2

9.Factorize the following expressions.

(a)m22m1(b)m24m4

(c)m28m16(d)m220m100

(e)2m28m8(f)5m24530m

10.Factorize the following expressions.

(a)a2bab25a5b(b)a2ab21b242b3

(c)2a49a28a2b36b(d)56ax4b232bx7ab

(e)b2x15b3abx245ax(f)24by3ax2bx36ay

11.(a)Factorize the following expressions.

(i)16x29y224xy(ii)4x228x49

(b)Using the results of (a), factorize(16x29y224xy)4(4x228x49).

12. (a)Factorize p24pq4q2.

(b)Hence factorize(m1)24(m1)(5n)4(5n)2.

Advanced Set

Level 1

1.Factorize the following expressions.

(a)4a8(b)3a9b

(c)27ab45(d)416x

(e)36x12y42(f)81x54y135z

2.Factorize the following expressions.

(a)40a20ab(b)32a27ab

(c)5xy10yz(d)30a29a2c2

(e)25x350x2y(f)7axy49a2x2y

3.Factorize the following expressions.

(a)a(b1)3(b1)(b)2x(y1)3(1y)

(c)5p(3q2)10(3q2)2(d)22yxxy

(e)2pr2psrs(f)axbxayby

4.Factorize the following expressions.

(a)x24(b)x281

(c)16a2b2(d)25x264y2

(e)8x250(f)4x281y4

Level 2

5.Factorize the following expressions.

(a)6axb3bx2a(b)6x23xz2xyyz

(c)10ax12by20bx6ay(d)20abx22016ax25bx

(e)12a2xy108ay15ax(f)18wx220yz230x2z212wy

6.Factorize the following expressions.

(a)200y232x2(b)81x2y16y(xy)2

(c)25681x4(d)(xy)4(xy)4

7.Factorize the following expressions.

(a)m26m9(b)8m16m21

(c)912m4m2(d)3m26m3

(e)8m240m50(f)18m224m8

8.Factorize the following expressions.

(a)4x29y212xy(b)a2b22abcc2

(c)36x2y22560xy(d)28a27b228ab

(e)12p227q236pq(f)36x24y2z224xyz

9.(a)Factorize the following expressions.

(i)16x224xy9y2(ii)z28z16

(b)Using the results of (a), factorize(16x224xy9y2)+8(4x3y)16.

10. (a)Factorize x4y4.

(b)Hence factorizex42x2zy42y2z.

11.(a)Factorize the following expressions.

(i)3y212y12(ii)y34y24y

(b)Using the results of (a), factorizey37y216y12.

12.Factorize the following expressions.

(a)(2xy)2y24x2(b)z2(y225x210xy)

(c)4827a236ab12b2(d)9w224wz16z236x236xy9y2

Exercise 3B

Elementary Set

Level 1

1.In each of the following, find the numbers p and q.

(a)pq2, pq3(b)pq8, pq6

(c)pq14, pq9(d)pq10, pq3

2.Kentfactorizesthequadratic polynomials by using cross multiplication. Fill in the blanks with suitable expressions, and find the correct pair of factors to factorize the following expressions.

(a)x25x6

Therefore, ( ) and ( ) are the pair of factors.
x25x6( )( )
(b)y27y18

Therefore, ( ) and ( ) are the pair of factors.

y27y18( )( )

(c)z22z15

Therefore, ( ) and ( ) are the pair of factors.

z22z15( )( )

3.Factorize the following expressions.

(a)a28a7(b)a214a13

(c)b212b20(d)b212b27

(e)c214c48(f)c216c55

4.Factorize the following expressions.

(a)x29x14(b)x210x24

(c)y213y22(d)y217y16

(e)z211z24(f)z225z100

5.Factorize the following expressions.

(a)x2x12(b)x22x3

(c)y23y10(d)y214y15

(e)z23z54(f)z210z144

6.Factorize the following expressions.

(a)p210p11(b)p24p21

(c)q27q60(d)q28q20

(e)r212r64(f)r219r150

Level 2

7.Factorize the following expressions.

(a)200m230m(b)13m14m2

(c)187mm2(d)n25n6

(e)42nn2(f)10nn275

8.Factorize the following expressions.

(a)a(a13)36(b)a24(a15)

(c)3(5a12)a2(d)(a13)(a1)49

(e)(a6)(a13)20(f)14(a15)(a10)

9.(a)If x2ax6(x2)(xb), find the values of a and b.

(b)Hence factorize (xb)(x5)20a.

Advanced Set

Level 1

1.In each of the following, find the numbers p and q.

(a)pq10, pq7(b)pq24, pq10

(c)pq14, pq5(d)pq24, pq5

2.In the diagram below, circle the factors of x27x30.

3.Factorize the following expressions.

(a)a24a3(b)a212a32

(c)b29b18(d)b27b10

(e)c223c120(f)c219c90

4.Factorize the following expressions.

(a)x24x5(b)x218x19

(c)y26y27(d)y2y2

(e)z22z24(f)z24z45

Level 2

5.Factorize the following expressions.

(a)6aa2(b)b210029b

(c)9c22c2(d)15x28x

(e)y2365y(f)6z72z2

6.Factorize the following expressions.

(a)14p40p2(b)30qq2

(c)2rr248(d)14s213s

(e)100t215t(f)64u216u

7.Factorize the following expressions.

(a)a(a12)35(b)a25(3a10)

(c)7(2b21)b2(d)(b7)(b8)12

(e)180(c6)(c9)(f)(5c)(4c)90

8.(a)Factorize the following expressions.

(i)y25y4

(ii)y213y12

(b)Using the results of (a), factorize(y35y24y)(2y226y24).

9.(a)If x2ax96(xb)(x12), find the values of a and b.

(b)Hence factorize (ay)(yb)75.

Exercise 3C

Elementary Set

Level 1

1.In the diagram below, circle the factors of 3x211x4.

2.In the diagram below, circle the factors of 2x27x6.

3.In the diagram below, circle the factors of 6x211x10.

4.Factorize the following expressions.

(a)2x2x1(b)5x24x1

(c)6y25y1(d)4y25y1

(e)10z27z1(f)27z212z1

5.Factorize the following expressions.

(a)2a23a2(b)3a28a5

(c)3b25b2(d)2b25b7

(e)5c212c7(f)11c214c3

6.Factorize the following expressions.

(a)2p29p9(b)4p27p3

(c)8q23q5(d)6q223q7

(e)10r23r(f)r215r2

7.Factorize the following expressions.

(a)2m25m12(b)5m227m18

(c)31n127n2(d)11n43n2

(e)3k224k45(f)203k24k

Level 2

8.Factorize the following expressions.

(a)9s25s4(b)6s211s21

(c)8t26t27(d)4t28t45

(e)10u221u9(f)12u23611u

9.Factorize the following expressions.

(a)6x23x9(b)12x232x20

(c)18y26y4(d)44102y20y2

(e)40z66z26(f)2z(4z1)10

10. (a)Factorize 6p213p5.

(b)Hence factorize6(2q3)213(2q3)5.

Advanced Set

Level 1

1.In the diagram below, circle the factors of 9x247x10.

2.In the diagram below, circle the factors of 16x218x9.

3.(a)Assume that2x2x5 can be factorized by using the cross multiplication, what should be the sum of the cross products?

(b)Fill in the blanks with suitable expressions.

For 2x2x5,

(c)Based on the result of (b), is the assumption in (a) correct?

4.Factorize the following expressions.

(a)5x214x3(b)7x236x5

(c)2a2a3(d)3p2118p

(e)7x274x(f)27a5a2

5.Factorize the following expressions.

(a)6p27p5(b)10a27a3

(c)13p310p2(d)34m24m

(e)10n312n2(f)25q18q2

6.Factorize the following expressions.

(a)4x220x24(b)3x221x30

(c)28x847x2(d)304x2x2

(e)12010x5x2(f)644x26x

Level 2

7.Factorize the following expressions.

(a)18a23a10(b)21y2y36

(c)24a23522a(d)4p29p6

(e)p2012p2(f)1810q257q

8.Factorize the following expressions.

(a)30x235x10(b)20x276x16

(c)15x1812x2(d)27p6p215

(e)4x(14x)30(f)216p(38p)

9.(a)Factorize (x1)(x6)8.

(b)Hence factorize(2y25y1)(2y25y6)8.

10. (a)Factorize 16y216y4.

(b)Hence, or otherwise, factorize(16x416x24)49x2.

Exercise 3D

Elementary Set

Level 1

1.Factorize the following quadratic polynomials in two variables by using the diagram below.

(a)x29xy14y2( )( )

(b)x215xy14y2( )( )

2.In the diagram below, circle the factors of 8x230xy25y2.

3.Factorize the following expressions.

(a)x2xy2y2(b)x24xy3y2

(c)5m24mnn2(d)m212mn13n2

(e)u211v210uv(f)3u2v22uv

4.Factorize the following expressions.

(a)a25ab6b2(b)a28ab9b2

(c)10p27pqq2(d)p225q210pq

(e)3rs10s2r2(f)8r2s22rs

5.Factorize the following expressions.

(a)x26xy16y2(b)18x29xyy2

(c)p211pq30q2(d)70p2q219pq

(e)14sts224t2(f)5stt236s2

Level 2

6.Factorize the following expressions.

(a)5a216ab11b2(b)2a221ab19b2

(c)24m22mn5n2(d)5m223mn12n2

(e)13uv3v230u2(f)15u28uv12v2

7.Factorize the following expressions.

(a)5x230xy35y2(b)15p26pq21q2

(c)14p249pq42q2(d)4x212xy40y2

(e)30ab4a2100b2(f)20a2105b240ab

8.(a)Factorize 6x213xy28y2.

(b)Hence factorize6x213xy28y210x35y.

Advanced Set

Level 1

1.In the diagram below, circle the factors of 6x2xy15y2.

2.Factorize the following expressions.

(a)m26mn5n2(b)3x22xyy2

(c)x25xy14y2(d)8a26abb2

(e)u217v216uv(f)p232q24pq

3.Factorize the following expressions.

(a)2x223xy11y2(b)5x28xy13y2

(c)2a211ab21b2(d)27p215pq2q2

(e)7a23b220ab(f)11pq10q23p2

4.Factorize the following expressions.

(a)11u219uv8v2(b)5u243uv18v2

(c)6m217mn12n2(d)24m27n238mn

(e)11r28s218rs(f)16r25s218rs

5.Factorize the following expressions.

(a)6a25ab4b2(b)8a218ab81b2

(c)20x211xy42y2(d)15y217yz4z2

(e)9u222uv15v2(f)7uv3v26u2

6.Factorize the following expressions.

(a)9xy54y23x2(b)100y25x25xy

(c)24(r22t2)88rt(d)10r234rt28t2

(e)24a220b228ab(f)18b216a224ab

Level 2

7.(a)Factorize the following expressions.

(i)6x2xy2y2(ii)2x2xy2xy

(b)Using the results of (a), factorize 10x2xy2y24x2y.

8.(a)Factorize the following expressions.

(i)4x212xy9y2(ii)9y224yz16z2

(b)Using the results of (a), factorize4(4x212xy9y2)25(9y224yz+16z2).

Exercise 3E

Elementary Set

Level 1

1.Fill in the blanks with ‘’ or ‘’ to make the equalities hold.

(a)a3b3(a□b)(a2□ab□b2)(b) a3□b3(a□b)(a2ab□b2)

2.Find the values of p and q in each of the following.

(a)x31728(x12)(x2pxq)(b) x3512(xp)(x2qx64)

(c)343x31(px1)(qx27x1)

3.Factorize the following expressions.

(a)x323(b)x353

(c)x3103

4.Factorize the following expressions.

(a)y3216(b)125y3

(c)y38

5.Factorize the following expressions.

(a)23x31(b)(15x)31

(c)216x31

6.Factorize the following expressions.

(a)x3(9y)3(b)(12x)3y3

(c)x327y3

7.Factorize the following expressions.

(a)(4y)3113(b)163(9y)3

(c)1000y3343

8.Factorize the following expressions.

(a)2a32b3(b)4a332b3

(c)54a3250b3

Level 2

9.Factorize the following expressions.

(a)ab3ac3(b)9a372

(c)6ab348ac3(d)a3b264b2c3

10.Factorize the following expressions.

(a)63(43x)3(b)(2x)3(2x)3

(c)125x3(xy)3(d)(5a6)3729

Advanced Set

Level 1

1.Factorize the following expressions.

(a)x363(b)53x31

(c)43x31

2.Factorize the following expressions.

(a)x38(b)x327

(c)1000x31

3.Factorize the following expressions.

(a)x3(20y)3(b)(11x)3y3

(c)125x3y3

4.Factorize the following expressions.

(a)(11x)393(b)64x327

(c)216125x3

5.Factorize the following expressions.

(a)4x34y3(b)5a340b3

(c)189s356t3

6.Factorize the following expressions.

(a)2x3128(b)50005x3

(c)48x36(d)81x324

Level 2

7.Factorize the following expressions.

(a)mn3mp3(b)2x2y316x5

(c)125p4qpq4(d)54u6v2250u3v5

8.Factorize the following expressions.

(a)153(x5)3(b)64y3(4y)3

(c)x3y3(3x2y3xy2)(d)(a2b)3(3ab)3

9.(a)Factorize a6b6.

[Hint: a6b6(a3)2(b3)2]

(b)Hence factorize (xy)6(xy)6.

10. (a)Factorize 8x327.

(b)Hence factorize 16x34x48.

Chapter Test / (Time allowed: 1 hour)

SectionA(1) [3 marks each]

1.Factorize the following expressions.

(a)30ab15a2b

(b)x214xy49y2

2.Factorize xy8y9x72.

3.Factorize 24t22t1.

4.Factorize 3u214u5.

5.Factorize 3a22abb2.

6.Factorize 127q3.

SectionA(2) [6 marks each]

7.Factorize 10x212x38x.

8.Factorize 16a215b28ab.

9.Factorize (x2)24(x24x12).

10.Factorize 27s3125t36s10t.

Section B

11.(a)Factorize 3x28x80.(4 marks)

(b)Factorize x364.(3 marks)

(c)Hence factorize x36x216x96.(6 marks)

Multiple Choice Questions [3 marks each]

Chapter 3More about Factorization 3.1

12.Which of the following must be correct?

I.a2b2(ab)(ab)

II.(ab)2a22abb2

III.(ab)2a22abb2

A.II only

B.I and III only

C.II and III only

D.I, II and III□

13.If ax2y2(2xy)(y2x), then a

A.4.

B.2.

C.2.

D.4.□

14.xyy1x

A.(x1)(y1).

B.(x1)(y1).

C.(1x)(y1).

D.2xy(1x).□

15.a2b3cbc3

A.a2b2c2.

B.bc3(a1)(a1).

C.bc(abc)(abc).

D.bc(abc)2.□

16.(x6)(7x)4x

A.(x3)(x14).

B.(x3)(x14).

C.(x3)(x14).

D.(x3)(x14). □

17.166aa2

A.(a4)2.

B.(a4)2.

C.(a8)(a2).

D.(a8)(a2).□

18.13x30x2

A.(x15)(x2).

B.(x15)(x2).

C.(x3)(x10).

D.(x3)(x10). □

19.16u260v216uv

A.(u12v)(16u5v).

B.2(u6v)(8u5v).

C.4(u5v)(4u3v).

D.4(2u3v)(2u5v).□

20.(x1)22x25x3

A.(x1)(x2).

B.(x1)(2x3).

C.(x1)2(2x3).

D.(x2)(2x3). □

21.Which of the following has the factor 2x1?

A.4x32x2

B.6x2x2

C.2x21

D.4xy2y6x3□

22.Which of the following does not have the factor 3x2?

A.3x24

B.9x24x

C.(9x34x)(18x12)

D.(9x34x)(18x12)□

23.It is known that 3x2ax6, where a is an integer, can be factorized by using the cross multiplication. Which of the following is/are not the possible value(s) of a?

I.7

II.3

III.9

A.II only

B.III only

C.I and III only

D.II and III only□

24.Which of the following must be correct?

I.a3b3(ab)(a22abb2)

II.a3b3(ab)(a2abb2)

III.a2b2(ab)(ab)

A.I only

B.II only

C.III only

D.None of the above□

25.8x3y3

A.(2xy)(4x22xyy2).

B.(2xy)(4x2xyy2).

C.(2xy)(4x22xyy2).

D.(2xy)(4x22xyy2).□

26.Which of the following have the factor 3xy?

I.3ax3byay9bx

II.27x3y3

III.27x3y3

A.I and II only

B.I and III only

C.II and III only

D.I, II and III□