X2 T02 03 roots & coefficients
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Transcript of X2 T02 03 roots & coefficients
![Page 1: X2 T02 03 roots & coefficients](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/1.jpg)
Roots and Coefficients
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Roots and Coefficients 21For nnn cxbxaxxP
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Roots and Coefficients 21For nnn cxbxaxxP
) i.e. timeaat 1 roots, of (sum
ab
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Roots and Coefficients 21For nnn cxbxaxxP
) i.e. timeaat 1 roots, of (sum
ab
) i.e. timeaat 2 roots, of (sum ac
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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](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/6.jpg)
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
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2222
We have already seen that for ; cbxax 2
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2222
11and
We have already seen that for ; cbxax 2
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2222
11and
We have already seen that for ; cbxax 2
This can be generalised to;
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2222
11and
222
We have already seen that for ; cbxax 2
This can be generalised to;
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2222
11and
222
2)(order 1
We have already seen that for ; cbxax 2
This can be generalised to;
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2222
11and
222
2)(order 1
We have already seen that for ; cbxax 2
This can be generalised to;
3)(order
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2222
11and
222
2)(order 1
We have already seen that for ; cbxax 2
This can be generalised to;
3)(order
4)(order
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e.g. (i) find; ,0432 23 xxx
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e.g. (i) find; ,0432 23 xxx
c)
b)
a)
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e.g. (i) find; ,0432 23 xxx
c)
b)
a) = 2
= 3
= 4
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e.g. (i) find; ,0432 23 xxx
c)
b)
a) = 2
= 3
= 4
2 d)
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e.g. (i) find; ,0432 23 xxx
c)
b)
a) = 2
= 3
= 4 22 2 d)
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e.g. (i) find; ,0432 23 xxx
c)
b)
a) = 2
= 3
= 4 22 2 d)
2
3222
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e.g. (i) find; ,0432 23 xxx
c)
b)
a) = 2
= 3
= 4 22 2 d)
2
3222
3 e)
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e.g. (i) find; ,0432 23 xxx
c)
b)
a) = 2
= 3
= 4 22 2 d)
2
3222
3 e) 0432 23
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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
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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
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01232 23
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01232 23
1232 23
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01232 23
1232 23
2
122322
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01232 23
1232 23
2
122322
4 f)
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01232 23
1232 23
2
122322
4 f) 0432 234
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01232 23
1232 23
2
122322
4 f) 0432 234
04320432
234
234
![Page 30: X2 T02 03 roots & coefficients](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/30.jpg)
01232 23
1232 23
2
122322
4 f) 0432 234
04320432
234
234
0432 234
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01232 23
1232 23
2
122322
4 f) 0432 234
04320432
234
234
0432 234
432 234
![Page 32: X2 T02 03 roots & coefficients](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/32.jpg)
01232 23
1232 23
2
122322
4 f) 0432 234
04320432
234
234
0432 234
432 234
18
242322
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g)
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g)
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g) 222
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g) 222
2482242448
2224
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g) 222
2482242448
2224
2
432248
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(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
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(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](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/40.jpg)
(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](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/41.jpg)
(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)
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(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
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(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](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/44.jpg)
(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](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/45.jpg)
(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](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/46.jpg)
(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
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c) Show that abcdac 22
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c) Show that abcdac 22
2 (2); Using kbq
![Page 49: X2 T02 03 roots & coefficients](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/49.jpg)
c) Show that abcdac 22
2 (2); Using kbq
dkbk 22 (4); into Sub
![Page 50: X2 T02 03 roots & coefficients](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/50.jpg)
c) Show that abcdac 22
2 (2); Using kbq
dkbk 22 (4); into Subdkbk 42
![Page 51: X2 T02 03 roots & coefficients](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/51.jpg)
c) Show that abcdac 22
2 (2); Using kbq
dkbk 22 (4); into Subdkbk 42
dac
abc
2
2
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c) Show that abcdac 22
2 (2); Using kbq
dkbk 22 (4); into Subdkbk 42
dac
abc
2
2
)ac( 2 k
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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](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/54.jpg)
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](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/55.jpg)
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](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/56.jpg)
d) If 2 is also a root of the equation, and b = 0, show that c is even.
root4th thebeLet
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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](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/58.jpg)
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](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/59.jpg)
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](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/60.jpg)
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](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/61.jpg)
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](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/62.jpg)
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](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/63.jpg)
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](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/64.jpg)
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](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/65.jpg)
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](https://reader033.fdocuments.us/reader033/viewer/2022051113/55b6026cbb61eb1c0a8b46fe/html5/thumbnails/66.jpg)
Exercise 5D; 1 to 9