Cambridge Additional Mathematics

474 Answers

EXERCISE 6A.3

1aquotient isx+1, remainder is¡x¡ 4

bquotient is 3 , remainder is¡x+3

c quotient is 3 x, remainder is¡ 2 x¡ 1

dquotient is 0 , remainder isx¡ 4

2a 1 ¡

2 x

x^2 +x+1

, x^2 ¡x+1=1(x^2 +x+1)¡ 2 x

bx¡

2 x

x^2 +2

,x^3 =x(x^2 +2)¡ 2 x

c x^2 +x+3+^3 x¡^4

x^2 ¡x+1

,

x^4 +3x^2 +x¡1=(x^2 +x+ 3)(x^2 ¡x+1)+3x¡ 4

d 2 x+4+^5 x+2

(x¡1)^2

,

2 x^3 ¡x+6=(2x+ 4)(x¡1)^2 +5x+2

ex^2 ¡ 2 x+3¡^4 x+3

(x+1)^2

,

x^4 =(x^2 ¡ 2 x+ 3)(x+1)^2 ¡ 4 x¡ 3

f x^2 ¡ 3 x+5+^15 ¡^10 x

(x¡1)(x+2)

,

x^4 ¡ 2 x^3 +x+5 = (x^2 ¡ 3 x+5)(x¡1)(x+2)+15¡ 10 x

3 quotient isx^2 +2x+3, remainder is 7

4 quotient isx^2 ¡ 3 x+5, remainder is 15 ¡ 10 x

EXERCISE 6B.1

1a 4 ,¡^32 b ¡ 3 §

p

10 c 5 §

p

19

d 0 ,§ 2 e 0 ,§

p

11 f § 2 ,§

p

2

2a 1 ,¡^25 b ¡^12 ,§

p

3 c ¡ 3 ,^13 , 2

d 0 , 1 §

p

3 e 0 ,§

p

7 f §

p

2 ,§

p

5

3a(2x+ 3)(x¡5) bx(x¡7)(x¡4)

c (x¡ 3 ¡

p

6)(x¡3+

p

6)

dx(x+1+

p

5)(x+1¡

p

5) ex(3x¡2)(2x+1)

f (x+ 1)(x¡1)(x+

p

5)(x¡

p

5)

4 P(®)=0, P( ̄)=0, P(°)=0

5aP(x)=a(x+ 3)(x¡4)(x¡5), a 6 =0

bP(x)=a(x+ 2)(x¡2)(x¡3), a 6 =0

c P(x)=a(x¡3)(x^2 ¡ 2 x¡4), a 6 =0

dP(x)=a(x+ 1)(x^2 +4x+2), a 6 =0

6aP(x)=a(x^2 ¡1)(x^2 ¡2), a 6 =0

bP(x)=a(x¡2)(5x+ 1)(x^2 ¡3), a 6 =0

c P(x)=a(x+ 3)(4x¡1)(x^2 ¡ 2 x¡1), a 6 =0

dP(x)=a(x^2 ¡ 4 x¡1)(x^2 +4x¡3), a 6 =0

EXERCISE 6B.2

1aa=2,b=5, c=5 ba=3,b=4, c=3

c a=2,b=¡ 5 ,c=4

2aa=2,b=¡ 2 or a=¡ 2 ,b=2

ba=3,b=¡ 1

3aa=1,b=6, c=¡ 7 b(x+ 3)(x+ 7)(x¡1)

4ap=2,q=7,r=5 bx=^12 ,¡ 1 ,¡^52

5aa=3,b=¡ 2 ,c=1

b 3 x^3 +10x^2 ¡ 7 x+4=(x+ 4)(3x^2 ¡ 2 x+1)

¢of 3 x^2 ¡ 2 x+1is¡ 8 ,

) the only real zero is¡ 4.

6aa=1,b=¡ 2 ,c=¡ 1 ,k=¡ 4

b ¡^23 , 1 §

p

2

7aa=¡ 2 , b=2 b¡ 1 §

p

3

8 a=¡ 11 , zeros are^32 ,

¡ 3 §

p

13

2

9aa=¡ 9 , b=¡ 1

b P(x)=0whenx=¡ 1 ,¡^12 , 2 , 4

10 Hint: Let x^3 +3x^2 ¡ 9 x+c=(x+a)^2 (x+b)

Whenc=5, the cubic is (x¡1)^2 (x+5).

Whenc=¡ 27 , the cubic is (x+3)^2 (x¡3).

EXERCISE 6C

1aP(x)=Q(x)(x¡2) + 7, P(x) divided byx¡ 2

leaves a remainder of 7.

b P(¡3) =¡ 8 , P(x) divided byx+3leaves a

remainder of¡ 8.

c P(5) = 11, P(x)=Q(x)(x¡5) + 11

2a 4 b ¡ 19 c 1 34

4aa=3 ba=2 5 a=¡ 5 ,b=6

6 a=¡ 5 , b=6 7 ¡ 7

8aP(x)=Q(x)(2x¡1) +R

P(^12 )=Q(^12 )(2£^12 ¡1) +R

=Q(^12 )£0+R

=R

bi¡ 3 ii 7 iii ¡ 7

9 a=3,b=10 10 a¡ 3 b 1

EXERCISE 6D

1afactor bnot a factor c factor dnot a factor

2ac=2 b c=¡ 2 c b=3

3 k=¡ 8 , P(x)=(x+ 2)(x¡2)(2x+1)

4ak=¡ 8 b P(x)=(x¡3)(3x^2 +x¡2)

c x=¡ 1 ,^23 , 3

5 a=7,b=¡ 14 6 a=3,b=2

7aa=7,b=¡ 6 b 60

c P(x)=(x+ 3)(2x^2 +3x¡2) d ¡ 3 ,¡ 2 ,^12

8aa=7,b=2 bx=¡ 2 §

p

6

9a iP(a)=0, x¡ais a factor

ii(x¡a)(x^2 +ax+a^2 )

biP(¡a)=0, x+ais a factor

ii(x+a)(x^2 ¡ax+a^2 )

10 a=2

EXERCISE 6E

1ax=1, 2 , 3 b x=¡ 1 , 2 f 2 is a double rootg

c x=1,¡ 1 ,¡ 2 d x=¡ 1 , 3 , 4 ex=¡ 5 ,¡ 4 , 4

f x=¡ 3 ,¡ 5 f¡ 5 is a double rootg

2ax=¡ 2 , 2 , 3 b x=¡ 3 ,¡ 2 , 6 c x=¡ 3 , 4 , 7

REVIEW SET 6A

1a 8 x^2 +6x+3 b 7 x^2 ¡ 9 x+9

c 15 x^4 +32x^3 +29x¡ 4

2aquotient=2x+5, remainder=3

b quotient=x^2 ¡ 4 x+2, remainder=¡ 5

3a^43 ,¡ 2 b¡ 4 §

p

5

cyan magenta yellow black

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100 IB HL OPT
Sets Relations Groups
Y:\HAESE\CAM4037\CamAdd_AN\474CamAdd_AN.cdr Tuesday, 8 April 2014 8:33:13 AM BRIAN