11 x1 t15 06 roots & coefficients (2013)
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Transcript of 11 x1 t15 06 roots & coefficients (2013)
![Page 1: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/1.jpg)
Roots and Coefficients
![Page 2: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/2.jpg)
Roots and CoefficientsQuadratics 2 0ax bx c
![Page 3: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/3.jpg)
Roots and CoefficientsQuadratics 2 0ax bx c
ba
![Page 4: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/4.jpg)
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
![Page 5: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/5.jpg)
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
Cubics 3 2 0ax bx cx d
![Page 6: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/6.jpg)
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
Cubics 3 2 0ax bx cx d
ba
![Page 7: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/7.jpg)
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
Cubics 3 2 0ax bx cx d
ba
ca
![Page 8: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/8.jpg)
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
Cubics 3 2 0ax bx cx d
ba
ca
da
![Page 9: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/9.jpg)
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
![Page 10: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/10.jpg)
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
![Page 11: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/11.jpg)
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
![Page 12: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/12.jpg)
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
![Page 13: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/13.jpg)
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
![Page 14: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/14.jpg)
For the polynomial equation;
0321 nnnn dxcxbxax
![Page 15: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/15.jpg)
For the polynomial equation;
ab
0321 nnnn dxcxbxax
(sum of roots, one at a time)
![Page 16: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/16.jpg)
For the polynomial equation;
ab
0321 nnnn dxcxbxax
(sum of roots, one at a time)
ac
(sum of roots, two at a time)
![Page 17: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/17.jpg)
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)
![Page 18: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/18.jpg)
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)
![Page 19: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/19.jpg)
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:
![Page 20: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/20.jpg)
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
![Page 21: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/21.jpg)
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
![Page 22: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/22.jpg)
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
![Page 23: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/23.jpg)
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
![Page 24: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/24.jpg)
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
![Page 25: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/25.jpg)
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
![Page 26: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/26.jpg)
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)
![Page 27: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/27.jpg)
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)
![Page 28: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/28.jpg)
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
![Page 29: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/29.jpg)
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
![Page 30: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/30.jpg)
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)
![Page 31: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/31.jpg)
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
![Page 32: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/32.jpg)
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
![Page 33: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/33.jpg)
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
![Page 34: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/34.jpg)
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
(i)
![Page 35: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/35.jpg)
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
0
(i)
![Page 36: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/36.jpg)
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
0
(i)
(ii)
![Page 37: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/37.jpg)
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
0
(i)
(ii)
1
![Page 38: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/38.jpg)
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
0
(i)
(ii)
1
111 (iii)
![Page 39: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/39.jpg)
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
0
(i)
(ii)
1
111 (iii)
111
![Page 40: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/40.jpg)
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
0
(i)
(ii)
1
111 (iii)
111
13
![Page 41: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/41.jpg)
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
0
3
(i)
(ii)
1
111 (iii)
111
13
![Page 42: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/42.jpg)
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
![Page 43: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/43.jpg)
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
![Page 44: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/44.jpg)
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
![Page 45: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/45.jpg)
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
![Page 46: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/46.jpg)
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
![Page 47: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/47.jpg)
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
![Page 48: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/48.jpg)
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
![Page 49: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/49.jpg)
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
![Page 50: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/50.jpg)
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
![Page 51: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/51.jpg)
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
![Page 52: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/52.jpg)
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
![Page 53: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/53.jpg)
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
![Page 54: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/54.jpg)
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
![Page 55: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/55.jpg)
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
![Page 56: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/56.jpg)
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
![Page 57: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/57.jpg)
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
![Page 58: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/58.jpg)
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
![Page 59: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/59.jpg)
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
![Page 60: 11 x1 t15 06 roots & coefficients (2013)](https://reader033.fdocuments.net/reader033/viewer/2022060116/557e78d3d8b42a03668b5164/html5/thumbnails/60.jpg)
Exercise 4F; 2, 4, 5ac, 6ac, 8, 10a, 13, 15, 16ad, 17, 18, 19