x2 t02 03 roots & coefficients (2012)

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Roots and Coefficients

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Roots and Coefficients

Roots and Coefficients 21For nnn cxbxaxxP

Roots and Coefficients 21For nnn cxbxaxxP

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

ab

Roots and Coefficients 21For nnn cxbxaxxP

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

ab

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

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

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

2222

We have already seen that for ; cbxax 2

2222

11and

We have already seen that for ; cbxax 2

2222

11and

We have already seen that for ; cbxax 2

This can be generalised to;

2222

11and

222

We have already seen that for ; cbxax 2

This can be generalised to;

2222

11and

222

2)(order 1

We have already seen that for ; cbxax 2

This can be generalised to;

2222

11and

222

2)(order 1

We have already seen that for ; cbxax 2

This can be generalised to;

3)(order

2222

11and

222

2)(order 1

We have already seen that for ; cbxax 2

This can be generalised to;

3)(order

4)(order

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

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

c)

b)

a)

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

c)

b)

a) = 2

= 3

= 4

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

c)

b)

a) = 2

= 3

= 4

2 d)

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

c)

b)

a) = 2

= 3

= 4 22 2 d)

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

c)

b)

a) = 2

= 3

= 4 22 2 d)

2

3222

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

c)

b)

a) = 2

= 3

= 4 22 2 d)

2

3222

3 e)

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

c)

b)

a) = 2

= 3

= 4 22 2 d)

2

3222

3 e) 0432 23

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

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

01232 23

01232 23

1232 23

01232 23

1232 23

2

122322

01232 23

1232 23

2

122322

4 f)

01232 23

1232 23

2

122322

4 f) 0432 234

01232 23

1232 23

2

122322

4 f) 0432 234

04320432

234

234

01232 23

1232 23

2

122322

4 f) 0432 234

04320432

234

234

0432 234

01232 23

1232 23

2

122322

4 f) 0432 234

04320432

234

234

0432 234

432 234

01232 23

1232 23

2

122322

4 f) 0432 234

04320432

234

234

0432 234

432 234

18

242322

g)

g)

g) 222

g) 222

2482242448

2224

g) 222

2482242448

2224

2

432248

(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

(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.

(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.

(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)

(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

(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

(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

(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

(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

c) Show that abcdac 22

c) Show that abcdac 22

2 (2); Using kbq

c) Show that abcdac 22

2 (2); Using kbq

dkbk 22 (4); into Sub

c) Show that abcdac 22

2 (2); Using kbq

dkbk 22 (4); into Subdkbk 42

c) Show that abcdac 22

2 (2); Using kbq

dkbk 22 (4); into Subdkbk 42

dac

abc

2

2

c) Show that abcdac 22

2 (2); Using kbq

dkbk 22 (4); into Subdkbk 42

dac

abc

2

2

)ac( 2 k

c) Show that abcdac 22

2 (2); Using kbq

dkbk 22 (4); into Subdkbk 42

dac

abc

2

2

)ac( 2 k

dacabc 22

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

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

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

root4th thebeLet

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

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

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

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

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

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

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

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

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

Exercise 5D; 1 to 9