Download - 11X1 T15 06 roots & coefficients (2011)
Roots and Coefficients
Roots and CoefficientsQuadratics 2 0ax bx c
Roots and CoefficientsQuadratics 2 0ax bx c
ba
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
Cubics 3 2 0ax bx cx d
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
Cubics 3 2 0ax bx cx d
ba
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
Cubics 3 2 0ax bx cx d
ba
ca
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
Cubics 3 2 0ax bx cx d
ba
ca
da
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
Cubics 3 2 0ax bx cx d
ba
ca
da
Quartics 4 3 2 0ax bx cx dx e
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
Cubics 3 2 0ax bx cx d
ba
ca
da
Quartics 4 3 2 0ax bx cx dx e
ba
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
Cubics 3 2 0ax bx cx d
ba
ca
da
Quartics 4 3 2 0ax bx cx dx e
ba
ca
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
Cubics 3 2 0ax bx cx d
ba
ca
da
Quartics 4 3 2 0ax bx cx dx e
ba
ca
da
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
Cubics 3 2 0ax bx cx d
ba
ca
da
Quartics 4 3 2 0ax bx cx dx e
ba
ca
da
ea
For the polynomial equation;
0321 nnnn dxcxbxax
For the polynomial equation;
ab
0321 nnnn dxcxbxax
(sum of roots, one at a time)
For the polynomial equation;
ab
0321 nnnn dxcxbxax
(sum of roots, one at a time)
ac
(sum of roots, two at a time)
For the polynomial equation;
ab
0321 nnnn dxcxbxax
(sum of roots, one at a time)
ac
(sum of roots, two at a time)
ad
(sum of roots, three at a time)
For the polynomial equation;
ab
0321 nnnn dxcxbxax
(sum of roots, one at a time)
ac
(sum of roots, two at a time)
ad
(sum of roots, three at a time)
ae
(sum of roots, four at a time)
For the polynomial equation;
ab
0321 nnnn dxcxbxax
222
(sum of roots, one at a time)
ac
(sum of roots, two at a time)
ad
(sum of roots, three at a time)
ae
(sum of roots, four at a time)
Note:
3 2e.g. ( ) If , and are the roots of 2 5 3 1 0, find the values of;
i x x x
a) 4 4 4 7
3 2e.g. ( ) If , and are the roots of 2 5 3 1 0, find the values of;
i x x x
a) 4 4 4 7
52
3 2e.g. ( ) If , and are the roots of 2 5 3 1 0, find the values of;
i x x x
a) 4 4 4 7
52
32
3 2e.g. ( ) If , and are the roots of 2 5 3 1 0, find the values of;
i x x x
a) 4 4 4 7
52
32
12
3 2e.g. ( ) If , and are the roots of 2 5 3 1 0, find the values of;
i x x x
a) 4 4 4 7
52
32
12
5 14 4 4 7 4 72 2
3 2e.g. ( ) If , and are the roots of 2 5 3 1 0, find the values of;
i x x x
a) 4 4 4 7
52
32
12
5 14 4 4 7 4 72 2
272
3 2e.g. ( ) If , and are the roots of 2 5 3 1 0, find the values of;
i x x x
a) 4 4 4 7
52
32
12
5 14 4 4 7 4 72 2
272
1 1 1b)
3 2e.g. ( ) If , and are the roots of 2 5 3 1 0, find the values of;
i x x x
a) 4 4 4 7
52
32
12
5 14 4 4 7 4 72 2
272
1 1 1b)
3 2e.g. ( ) If , and are the roots of 2 5 3 1 0, find the values of;
i x x x
a) 4 4 4 7
52
32
12
5 14 4 4 7 4 72 2
272
1 1 1b)
3212
3 2e.g. ( ) If , and are the roots of 2 5 3 1 0, find the values of;
i x x x
a) 4 4 4 7
52
32
12
5 14 4 4 7 4 72 2
272
1 1 1b)
3212
3
3 2e.g. ( ) If , and are the roots of 2 5 3 1 0, find the values of;
i x x x
a) 4 4 4 7
52
32
12
5 14 4 4 7 4 72 2
272
1 1 1b)
3212
3
2 2 2c)
3 2e.g. ( ) If , and are the roots of 2 5 3 1 0, find the values of;
i x x x
a) 4 4 4 7
52
32
12
5 14 4 4 7 4 72 2
272
1 1 1b)
3212
3
2 2 2c) 2 2
3 2e.g. ( ) If , and are the roots of 2 5 3 1 0, find the values of;
i x x x
a) 4 4 4 7
52
32
12
5 14 4 4 7 4 72 2
272
1 1 1b)
3212
3
2 2 2c) 2 2
25 322 2
3 2e.g. ( ) If , and are the roots of 2 5 3 1 0, find the values of;
i x x x
a) 4 4 4 7
52
32
12
5 14 4 4 7 4 72 2
272
1 1 1b)
3212
3
2 2 2c) 2 2
25 322 2
374
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
(i)
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
0
(i)
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
0
(i)
(ii)
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
0
(i)
(ii)
1
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
0
(i)
(ii)
1
111 (iii)
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
0
(i)
(ii)
1
111 (iii)
111
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
0
(i)
(ii)
1
111 (iii)
111
13
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
0
3
(i)
(ii)
1
111 (iii)
111
13
2003 Extension 1 HSC Q4c)It is known that two of the roots of the equation are reciprocals of each other.Find the value of k.
062 23 kxxx
2003 Extension 1 HSC Q4c)It is known that two of the roots of the equation are reciprocals of each other.Find the value of k.
062 23 kxxx
and 1, be roots Let the
2003 Extension 1 HSC Q4c)It is known that two of the roots of the equation are reciprocals of each other.Find the value of k.
062 23 kxxx
and 1, be roots Let the
261
2003 Extension 1 HSC Q4c)It is known that two of the roots of the equation are reciprocals of each other.Find the value of k.
062 23 kxxx
and 1, be roots Let the
261
3
2003 Extension 1 HSC Q4c)It is known that two of the roots of the equation are reciprocals of each other.Find the value of k.
062 23 kxxx
and 1, be roots Let the
261
3
03 P
2003 Extension 1 HSC Q4c)It is known that two of the roots of the equation are reciprocals of each other.Find the value of k.
062 23 kxxx
and 1, be roots Let the
261
3
03 P
063332 23 k
2003 Extension 1 HSC Q4c)It is known that two of the roots of the equation are reciprocals of each other.Find the value of k.
062 23 kxxx
and 1, be roots Let the
261
3
03 P
063332 23 k
063954 k
2003 Extension 1 HSC Q4c)It is known that two of the roots of the equation are reciprocals of each other.Find the value of k.
062 23 kxxx
and 1, be roots Let the
261
3
03 P
063332 23 k
063954 k393 k
2003 Extension 1 HSC Q4c)It is known that two of the roots of the equation are reciprocals of each other.Find the value of k.
062 23 kxxx
and 1, be roots Let the
261
3
03 P
063332 23 k
063954 k393 k13k
2006 Extension 1 HSC Q4a)The cubic polynomial , where r, s and t are real numbers, has three real zeros, 1, and
(i) Find the value of r
tsxrxxxP 23
2006 Extension 1 HSC Q4a)The cubic polynomial , where r, s and t are real numbers, has three real zeros, 1, and
r 1(i) Find the value of r
tsxrxxxP 23
2006 Extension 1 HSC Q4a)The cubic polynomial , where r, s and t are real numbers, has three real zeros, 1, and
r 11r
(i) Find the value of r
tsxrxxxP 23
2006 Extension 1 HSC Q4a)The cubic polynomial , where r, s and t are real numbers, has three real zeros, 1, and
r 11r
(i) Find the value of r
(ii) Find the value of s + t
tsxrxxxP 23
2006 Extension 1 HSC Q4a)The cubic polynomial , where r, s and t are real numbers, has three real zeros, 1, and
r 11r
s 11
(i) Find the value of r
(ii) Find the value of s + t
tsxrxxxP 23
2006 Extension 1 HSC Q4a)The cubic polynomial , where r, s and t are real numbers, has three real zeros, 1, and
r 11r
s 11
(i) Find the value of r
(ii) Find the value of s + t
2s
tsxrxxxP 23
2006 Extension 1 HSC Q4a)The cubic polynomial , where r, s and t are real numbers, has three real zeros, 1, and
r 11r
s 11
(i) Find the value of r
(ii) Find the value of s + t
2s t1
tsxrxxxP 23
2006 Extension 1 HSC Q4a)The cubic polynomial , where r, s and t are real numbers, has three real zeros, 1, and
r 11r
s 11
(i) Find the value of r
(ii) Find the value of s + t
2s t1
tsxrxxxP 23
2t
2006 Extension 1 HSC Q4a)The cubic polynomial , where r, s and t are real numbers, has three real zeros, 1, and
r 11r
s 11
(i) Find the value of r
(ii) Find the value of s + t
2s t1
0 ts
tsxrxxxP 23
2t
Exercise 4F; 2, 4, 5ac, 6ac, 8, 10a, 13, 15, 16ad, 17, 18, 19
