x2 t02 03 roots & coefficients (2011)

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Roots and Coefficients

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Page 1: X2 T02 03 roots & coefficients (2011)

Roots and Coefficients

Page 2: X2 T02 03 roots & coefficients (2011)

Roots and Coefficients 21For nnn cxbxaxxP

Page 3: X2 T02 03 roots & coefficients (2011)

Roots and Coefficients 21For nnn cxbxaxxP

) i.e. timeaat 1 roots, of (sum

ab

Page 4: X2 T02 03 roots & coefficients (2011)

Roots and Coefficients 21For nnn cxbxaxxP

) i.e. timeaat 1 roots, of (sum

ab

) i.e. timeaat 2 roots, of (sum ac

Page 5: X2 T02 03 roots & coefficients (2011)

Roots and Coefficients 21For nnn cxbxaxxP

) i.e. timeaat 1 roots, of (sum

ab

) i.e. timeaat 2 roots, of (sum ac

) i.e. timeaat 3 roots, of (sum

ad

Page 6: X2 T02 03 roots & coefficients (2011)

Roots and Coefficients 21For nnn cxbxaxxP

) i.e. timeaat 1 roots, of (sum

ab

) i.e. timeaat 2 roots, of (sum ac

) i.e. timeaat 3 roots, of (sum

ad

) timeaat 4 roots, of (sum ae

Page 7: X2 T02 03 roots & coefficients (2011)

2222

We have already seen that for ; cbxax 2

Page 8: X2 T02 03 roots & coefficients (2011)

2222

11and

We have already seen that for ; cbxax 2

Page 9: X2 T02 03 roots & coefficients (2011)

2222

11and

We have already seen that for ; cbxax 2

This can be generalised to;

Page 10: X2 T02 03 roots & coefficients (2011)

2222

11and

222

We have already seen that for ; cbxax 2

This can be generalised to;

Page 11: X2 T02 03 roots & coefficients (2011)

2222

11and

222

2)(order 1

We have already seen that for ; cbxax 2

This can be generalised to;

Page 12: X2 T02 03 roots & coefficients (2011)

2222

11and

222

2)(order 1

We have already seen that for ; cbxax 2

This can be generalised to;

3)(order

Page 13: X2 T02 03 roots & coefficients (2011)

2222

11and

222

2)(order 1

We have already seen that for ; cbxax 2

This can be generalised to;

3)(order

4)(order

Page 14: X2 T02 03 roots & coefficients (2011)

e.g. (i) find; ,0432 23 xxx

Page 15: X2 T02 03 roots & coefficients (2011)

e.g. (i) find; ,0432 23 xxx

c)

b)

a)

Page 16: X2 T02 03 roots & coefficients (2011)

e.g. (i) find; ,0432 23 xxx

c)

b)

a) = 2

= 3

= 4

Page 17: X2 T02 03 roots & coefficients (2011)

e.g. (i) find; ,0432 23 xxx

c)

b)

a) = 2

= 3

= 4

2 d)

Page 18: X2 T02 03 roots & coefficients (2011)

e.g. (i) find; ,0432 23 xxx

c)

b)

a) = 2

= 3

= 4 22 2 d)

Page 19: X2 T02 03 roots & coefficients (2011)

e.g. (i) find; ,0432 23 xxx

c)

b)

a) = 2

= 3

= 4 22 2 d)

2

3222

Page 20: X2 T02 03 roots & coefficients (2011)

e.g. (i) find; ,0432 23 xxx

c)

b)

a) = 2

= 3

= 4 22 2 d)

2

3222

3 e)

Page 21: X2 T02 03 roots & coefficients (2011)

e.g. (i) find; ,0432 23 xxx

c)

b)

a) = 2

= 3

= 4 22 2 d)

2

3222

3 e) 0432 23

Page 22: X2 T02 03 roots & coefficients (2011)

e.g. (i) find; ,0432 23 xxx

c)

b)

a) = 2

= 3

= 4 22 2 d)

2

3222

3 e) 0432 23

04320432

23

23

Page 23: X2 T02 03 roots & coefficients (2011)

e.g. (i) find; ,0432 23 xxx

c)

b)

a) = 2

= 3

= 4 22 2 d)

2

3222

3 e) 0432 23

04320432

23

23

01232 23

Page 24: X2 T02 03 roots & coefficients (2011)

01232 23

Page 25: X2 T02 03 roots & coefficients (2011)

01232 23

1232 23

Page 26: X2 T02 03 roots & coefficients (2011)

01232 23

1232 23

2

122322

Page 27: X2 T02 03 roots & coefficients (2011)

01232 23

1232 23

2

122322

4 f)

Page 28: X2 T02 03 roots & coefficients (2011)

01232 23

1232 23

2

122322

4 f) 0432 234

Page 29: X2 T02 03 roots & coefficients (2011)

01232 23

1232 23

2

122322

4 f) 0432 234

04320432

234

234

Page 30: X2 T02 03 roots & coefficients (2011)

01232 23

1232 23

2

122322

4 f) 0432 234

04320432

234

234

0432 234

Page 31: X2 T02 03 roots & coefficients (2011)

01232 23

1232 23

2

122322

4 f) 0432 234

04320432

234

234

0432 234

432 234

Page 32: X2 T02 03 roots & coefficients (2011)

01232 23

1232 23

2

122322

4 f) 0432 234

04320432

234

234

0432 234

432 234

18

242322

Page 33: X2 T02 03 roots & coefficients (2011)

g)

Page 34: X2 T02 03 roots & coefficients (2011)

g)

Page 35: X2 T02 03 roots & coefficients (2011)

g) 222

Page 36: X2 T02 03 roots & coefficients (2011)

g) 222

2482242448

2224

Page 37: X2 T02 03 roots & coefficients (2011)

g) 222

2482242448

2224

2

432248

Page 38: X2 T02 03 roots & coefficients (2011)

(ii) (1996)Consider the polynomial equation where a, b, c and d are all integers.Suppose the equation has a root of the form ki, where k is real and

0234 dcxbxaxx

0k

Page 39: X2 T02 03 roots & coefficients (2011)

(ii) (1996)Consider the polynomial equation where a, b, c and d are all integers.Suppose the equation has a root of the form ki, where k is real and

0234 dcxbxaxx

0ka) State why the conjugate –ki is also a root.

Page 40: X2 T02 03 roots & coefficients (2011)

(ii) (1996)Consider the polynomial equation where a, b, c and d are all integers.Suppose the equation has a root of the form ki, where k is real and

0234 dcxbxaxx

0ka) State why the conjugate –ki is also a root.

Roots appear in conjugate pairs when the coefficients are real.

Page 41: X2 T02 03 roots & coefficients (2011)

(ii) (1996)Consider the polynomial equation where a, b, c and d are all integers.Suppose the equation has a root of the form ki, where k is real and

0234 dcxbxaxx

0ka) State why the conjugate –ki is also a root.

Roots appear in conjugate pairs when the coefficients are real.

akc 2 that Show b)

Page 42: X2 T02 03 roots & coefficients (2011)

(ii) (1996)Consider the polynomial equation where a, b, c and d are all integers.Suppose the equation has a root of the form ki, where k is real and

0234 dcxbxaxx

0ka) State why the conjugate –ki is also a root.

Roots appear in conjugate pairs when the coefficients are real.

akc 2 that Show b) qpxxkixkixdcxbxaxx 2234

Page 43: X2 T02 03 roots & coefficients (2011)

(ii) (1996)Consider the polynomial equation where a, b, c and d are all integers.Suppose the equation has a root of the form ki, where k is real and

0234 dcxbxaxx

0ka) State why the conjugate –ki is also a root.

Roots appear in conjugate pairs when the coefficients are real.

akc 2 that Show b) qpxxkixkixdcxbxaxx 2234

qkpxkxkqpxxqkpxkxkqxpxx

qpxxkx

222234

2222234

222

Page 44: X2 T02 03 roots & coefficients (2011)

(ii) (1996)Consider the polynomial equation where a, b, c and d are all integers.Suppose the equation has a root of the form ki, where k is real and

0234 dcxbxaxx

0ka) State why the conjugate –ki is also a root.

Roots appear in conjugate pairs when the coefficients are real.

akc 2 that Show b) qpxxkixkixdcxbxaxx 2234

qkpxkxkqpxxqkpxkxkqxpxx

qpxxkx

222234

2222234

222

)4( )3( )2( )1( 222 dqkcpkbkqap

Page 45: X2 T02 03 roots & coefficients (2011)

(ii) (1996)Consider the polynomial equation where a, b, c and d are all integers.Suppose the equation has a root of the form ki, where k is real and

0234 dcxbxaxx

0ka) State why the conjugate –ki is also a root.

Roots appear in conjugate pairs when the coefficients are real.

akc 2 that Show b) qpxxkixkixdcxbxaxx 2234

qkpxkxkqpxxqkpxkxkqxpxx

qpxxkx

222234

2222234

222

)4( )3( )2( )1( 222 dqkcpkbkqap

(3)into(1)Substitute

Page 46: X2 T02 03 roots & coefficients (2011)

(ii) (1996)Consider the polynomial equation where a, b, c and d are all integers.Suppose the equation has a root of the form ki, where k is real and

0234 dcxbxaxx

0ka) State why the conjugate –ki is also a root.

Roots appear in conjugate pairs when the coefficients are real.

akc 2 that Show b) qpxxkixkixdcxbxaxx 2234

qkpxkxkqpxxqkpxkxkqxpxx

qpxxkx

222234

2222234

222

)4( )3( )2( )1( 222 dqkcpkbkqap

(3)into(1)Substitute cak 2

Page 47: X2 T02 03 roots & coefficients (2011)

c) Show that abcdac 22

Page 48: X2 T02 03 roots & coefficients (2011)

c) Show that abcdac 22

2 (2); Using kbq

Page 49: X2 T02 03 roots & coefficients (2011)

c) Show that abcdac 22

2 (2); Using kbq

dkbk 22 (4); into Sub

Page 50: X2 T02 03 roots & coefficients (2011)

c) Show that abcdac 22

2 (2); Using kbq

dkbk 22 (4); into Subdkbk 42

Page 51: X2 T02 03 roots & coefficients (2011)

c) Show that abcdac 22

2 (2); Using kbq

dkbk 22 (4); into Subdkbk 42

dac

abc

2

2

Page 52: X2 T02 03 roots & coefficients (2011)

c) Show that abcdac 22

2 (2); Using kbq

dkbk 22 (4); into Subdkbk 42

dac

abc

2

2

)ac( 2 k

Page 53: X2 T02 03 roots & coefficients (2011)

c) Show that abcdac 22

2 (2); Using kbq

dkbk 22 (4); into Subdkbk 42

dac

abc

2

2

)ac( 2 k

dacabc 22

Page 54: X2 T02 03 roots & coefficients (2011)

c) Show that abcdac 22

2 (2); Using kbq

dkbk 22 (4); into Subdkbk 42

dac

abc

2

2

)ac( 2 k

dacabc 22

abcdac 22

Page 55: X2 T02 03 roots & coefficients (2011)

d) If 2 is also a root of the equation, and b = 0, show that c is even.

Page 56: X2 T02 03 roots & coefficients (2011)

d) If 2 is also a root of the equation, and b = 0, show that c is even.

root4th thebeLet

Page 57: X2 T02 03 roots & coefficients (2011)

d) If 2 is also a root of the equation, and b = 0, show that c is even.

root4th thebeLet

integeran is integer;an is roots the

ofproduct theandmultiples,integer arerootseother thre theAs

Page 58: X2 T02 03 roots & coefficients (2011)

d) If 2 is also a root of the equation, and b = 0, show that c is even.

root4th thebeLet

integeran is integer;an is roots the

ofproduct theandmultiples,integer arerootseother thre theAs

bkikikikikiki 222

Page 59: X2 T02 03 roots & coefficients (2011)

d) If 2 is also a root of the equation, and b = 0, show that c is even.

root4th thebeLet

integeran is integer;an is roots the

ofproduct theandmultiples,integer arerootseother thre theAs

bkikikikikiki 222

Page 60: X2 T02 03 roots & coefficients (2011)

d) If 2 is also a root of the equation, and b = 0, show that c is even.

root4th thebeLet

integeran is integer;an is roots the

ofproduct theandmultiples,integer arerootseother thre theAs

bkikikikikiki 222

202

2

2

kk

Page 61: X2 T02 03 roots & coefficients (2011)

d) If 2 is also a root of the equation, and b = 0, show that c is even.

root4th thebeLet

integeran is integer;an is roots the

ofproduct theandmultiples,integer arerootseother thre theAs

bkikikikikiki 222

202

2

2

kk

akc 2c) From

Page 62: X2 T02 03 roots & coefficients (2011)

d) If 2 is also a root of the equation, and b = 0, show that c is even.

root4th thebeLet

integeran is integer;an is roots the

ofproduct theandmultiples,integer arerootseother thre theAs

bkikikikikiki 222

202

2

2

kk

akc 2c) From ac 2

Page 63: X2 T02 03 roots & coefficients (2011)

d) If 2 is also a root of the equation, and b = 0, show that c is even.

root4th thebeLet

integeran is integer;an is roots the

ofproduct theandmultiples,integer arerootseother thre theAs

bkikikikikiki 222

202

2

2

kk

akc 2c) From ac 2

Mc 2

Page 64: X2 T02 03 roots & coefficients (2011)

d) If 2 is also a root of the equation, and b = 0, show that c is even.

root4th thebeLet

integeran is integer;an is roots the

ofproduct theandmultiples,integer arerootseother thre theAs

bkikikikikiki 222

202

2

2

kk

akc 2c) From ac 2

Mc 2 integeran isintegers,both areand Ma

Page 65: X2 T02 03 roots & coefficients (2011)

d) If 2 is also a root of the equation, and b = 0, show that c is even.

root4th thebeLet

integeran is integer;an is roots the

ofproduct theandmultiples,integer arerootseother thre theAs

bkikikikikiki 222

202

2

2

kk

akc 2c) From ac 2

Mc 2 integeran isintegers,both areand Ma

Thus c is an even number, as it is divisible by 2

Page 66: X2 T02 03 roots & coefficients (2011)

Exercise 5D; 1 to 9