11x1 t16 06 roots & coefficients
TRANSCRIPT
![Page 1: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/html5/thumbnails/1.jpg)
Roots and Coefficients
![Page 2: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/html5/thumbnails/2.jpg)
Roots and CoefficientsQuadratics 2 0ax bx c
![Page 3: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/html5/thumbnails/3.jpg)
Roots and CoefficientsQuadratics 2 0ax bx c
ba
![Page 4: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/html5/thumbnails/4.jpg)
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
![Page 5: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/html5/thumbnails/5.jpg)
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
Cubics 3 2 0ax bx cx d
![Page 6: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/html5/thumbnails/6.jpg)
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
Cubics 3 2 0ax bx cx d
ba
![Page 7: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/html5/thumbnails/7.jpg)
Roots and CoefficientsQuadratics 2 0ax bx c
ba
ca
Cubics 3 2 0ax bx cx d
ba
ca
![Page 8: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/html5/thumbnails/14.jpg)
For the polynomial equation;
0321 nnnn dxcxbxax
![Page 15: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/html5/thumbnails/15.jpg)
For the polynomial equation;
ab
0321 nnnn dxcxbxax
(sum of roots, one at a time)
![Page 16: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/html5/thumbnails/34.jpg)
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
(i)
![Page 35: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/html5/thumbnails/35.jpg)
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
0
(i)
![Page 36: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/html5/thumbnails/36.jpg)
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
0
(i)
(ii)
![Page 37: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/html5/thumbnails/37.jpg)
1988 Extension 1 HSC Q2c)If are the roots of find:0133 xx and,
0
(i)
(ii)
1
![Page 38: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/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: 11X1 T16 06 roots & coefficients](https://reader034.vdocuments.net/reader034/viewer/2022052622/5591372e1a28ab07498b45b7/html5/thumbnails/60.jpg)
Exercise 4F; 2, 4, 5ac, 6ac, 8, 10a, 13, 15, 16ad, 17, 18, 19