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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