11X1 T13 03 polynomial division
-
Upload
nigel-simmons -
Category
Education
-
view
382 -
download
1
Transcript of 11X1 T13 03 polynomial division
![Page 1: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/1.jpg)
Polynomial Division
![Page 2: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/2.jpg)
Polynomial Division xRxQxAxP
where;
![Page 3: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/3.jpg)
Polynomial Division xRxQxAxP
where;A(x) is the divisor
![Page 4: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/4.jpg)
Polynomial Division xRxQxAxP
where;A(x) is the divisor Q(x) is the quotient
![Page 5: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/5.jpg)
Polynomial Division xRxQxAxP
where;A(x) is the divisor Q(x) is the quotientR(x) is the remainder
![Page 6: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/6.jpg)
Polynomial Division xRxQxAxP
where;A(x) is the divisor
degree R(x) < degree A(x)
Q(x) is the quotientR(x) is the remainder
Note:
![Page 7: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/7.jpg)
Polynomial Division xRxQxAxP
where;A(x) is the divisor
degree R(x) < degree A(x)
Q(x) is the quotientR(x) is the remainder
Note:
Q(x) and R(x) are unique
![Page 8: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/8.jpg)
Find the quotient, Q(x), and the remainder, R(x), when the polynomial is divided by
Polynomial Division xRxQxAxP
where;A(x) is the divisor
degree R(x) < degree A(x)
124 xxxP
Q(x) is the quotientR(x) is the remainder
Note:
Q(x) and R(x) are unique
1998 Extension 1 HSC Q2a)
12 x
![Page 9: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/9.jpg)
Find the quotient, Q(x), and the remainder, R(x), when the polynomial is divided by
Polynomial Division xRxQxAxP
where;A(x) is the divisor
degree R(x) < degree A(x)
124 xxxP
Q(x) is the quotientR(x) is the remainder
Note:
Q(x) and R(x) are unique
1998 Extension 1 HSC Q2a)
12 x
1001 2342 xxxxx
![Page 10: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/10.jpg)
Find the quotient, Q(x), and the remainder, R(x), when the polynomial is divided by
Polynomial Division xRxQxAxP
where;A(x) is the divisor
degree R(x) < degree A(x)
124 xxxP
Q(x) is the quotientR(x) is the remainder
Note:
Q(x) and R(x) are unique
1998 Extension 1 HSC Q2a)
12 x
1001 2342 xxxxx
2x
![Page 11: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/11.jpg)
Find the quotient, Q(x), and the remainder, R(x), when the polynomial is divided by
Polynomial Division xRxQxAxP
where;A(x) is the divisor
degree R(x) < degree A(x)
124 xxxP
Q(x) is the quotientR(x) is the remainder
Note:
Q(x) and R(x) are unique
1998 Extension 1 HSC Q2a)
12 x
1001 2342 xxxxx
2x
234 0 xxx
![Page 12: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/12.jpg)
Find the quotient, Q(x), and the remainder, R(x), when the polynomial is divided by
Polynomial Division xRxQxAxP
where;A(x) is the divisor
degree R(x) < degree A(x)
124 xxxP
Q(x) is the quotientR(x) is the remainder
Note:
Q(x) and R(x) are unique
1998 Extension 1 HSC Q2a)
12 x
1001 2342 xxxxx
2x
234 0 xxx 22x
![Page 13: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/13.jpg)
Find the quotient, Q(x), and the remainder, R(x), when the polynomial is divided by
Polynomial Division xRxQxAxP
where;A(x) is the divisor
degree R(x) < degree A(x)
124 xxxP
Q(x) is the quotientR(x) is the remainder
Note:
Q(x) and R(x) are unique
1998 Extension 1 HSC Q2a)
12 x
1001 2342 xxxxx
2x
234 0 xxx 22x 10 x
![Page 14: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/14.jpg)
Find the quotient, Q(x), and the remainder, R(x), when the polynomial is divided by
Polynomial Division xRxQxAxP
where;A(x) is the divisor
degree R(x) < degree A(x)
124 xxxP
Q(x) is the quotientR(x) is the remainder
Note:
Q(x) and R(x) are unique
1998 Extension 1 HSC Q2a)
12 x
1001 2342 xxxxx
2x
234 0 xxx
2
22x 10 x
![Page 15: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/15.jpg)
Find the quotient, Q(x), and the remainder, R(x), when the polynomial is divided by
Polynomial Division xRxQxAxP
where;A(x) is the divisor
degree R(x) < degree A(x)
124 xxxP
Q(x) is the quotientR(x) is the remainder
Note:
Q(x) and R(x) are unique
1998 Extension 1 HSC Q2a)
12 x
1001 2342 xxxxx
2x
234 0 xxx
2
22x 10 x202 2 xx
![Page 16: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/16.jpg)
Find the quotient, Q(x), and the remainder, R(x), when the polynomial is divided by
Polynomial Division xRxQxAxP
where;A(x) is the divisor
degree R(x) < degree A(x)
124 xxxP
Q(x) is the quotientR(x) is the remainder
Note:
Q(x) and R(x) are unique
1998 Extension 1 HSC Q2a)
12 x
1001 2342 xxxxx
2x
234 0 xxx
2
22x
3
10 x202 2 xx
![Page 17: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/17.jpg)
Find the quotient, Q(x), and the remainder, R(x), when the polynomial is divided by
Polynomial Division xRxQxAxP
where;A(x) is the divisor
degree R(x) < degree A(x)
124 xxxP
Q(x) is the quotientR(x) is the remainder
Note:
Q(x) and R(x) are unique
1998 Extension 1 HSC Q2a)
12 x
1001 2342 xxxxx
2x
234 0 xxx
2
22x
3
10 x202 2 xx 3 and 22 xRxxQ
![Page 18: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/18.jpg)
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
![Page 19: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/19.jpg)
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
321210213 2342 xxxxxx
![Page 20: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/20.jpg)
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
321210213 2342 xxxxxx
22x
![Page 21: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/21.jpg)
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
321210213 2342 xxxxxx
22x
234 262 xxx
![Page 22: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/22.jpg)
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
321210213 2342 xxxxxx
22x
234 262 xxx 23 104 xx
![Page 23: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/23.jpg)
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
321210213 2342 xxxxxx
22x
234 262 xxx 23 104 xx 32 x
![Page 24: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/24.jpg)
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
321210213 2342 xxxxxx
22x
234 262 xxx
x4
23 104 xx 32 x
![Page 25: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/25.jpg)
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
321210213 2342 xxxxxx
22x
234 262 xxx
x4
23 104 xx 32 xxxx 4124 23
![Page 26: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/26.jpg)
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
321210213 2342 xxxxxx
22x
234 262 xxx
x4
23 104 xx
xx 62 2
32 xxxx 4124 23
![Page 27: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/27.jpg)
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
321210213 2342 xxxxxx
22x
234 262 xxx
x4
23 104 xx
xx 62 2
32 xxxx 4124 23
3
![Page 28: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/28.jpg)
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
321210213 2342 xxxxxx
22x
234 262 xxx
x4
23 104 xx
xx 62 2
32 xxxx 4124 23
3
2
![Page 29: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/29.jpg)
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
321210213 2342 xxxxxx
22x
234 262 xxx
x4
23 104 xx
xx 62 2
32 xxxx 4124 23
3
2
262 2 xx
![Page 30: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/30.jpg)
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
321210213 2342 xxxxxx
22x
234 262 xxx
x4
23 104 xx
xx 62 2
32 xxxx 4124 23
3
2
262 2 xx1
![Page 31: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/31.jpg)
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
321210213 2342 xxxxxx
22x
234 262 xxx
x4
23 104 xx
xx 62 2
32 xxxx 4124 23
3
2
262 2 xx1
(ii) Hence write f(x) = g(x)q(x) + r(x) where q(x) and r(x) are polynomials and r(x) has degree less than 2.
![Page 32: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/32.jpg)
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
321210213 2342 xxxxxx
22x
234 262 xxx
x4
23 104 xx
xx 62 2
32 xxxx 4124 23
3
2
262 2 xx1
(ii) Hence write f(x) = g(x)q(x) + r(x) where q(x) and r(x) are polynomials and r(x) has degree less than 2.
1242133212102 22234 xxxxxxxx
![Page 33: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/33.jpg)
Further graphing of polynomials
xAxRxQ
xAxPy
![Page 34: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/34.jpg)
Further graphing of polynomials
xAxRxQ
xAxPy
solve A(x) = 0 to find vertical asymptotes
![Page 35: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/35.jpg)
Further graphing of polynomials
xAxRxQ
xAxPy
solve A(x) = 0 to find vertical asymptotes
y = Q(x) is the horizontal/oblique
asymptote
![Page 36: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/36.jpg)
Further graphing of polynomials
xAxRxQ
xAxPy
solve A(x) = 0 to find vertical asymptotes
y = Q(x) is the horizontal/oblique
asymptote
solve R(x) = 0 to find where (if anywhere) the curve cuts
the horizontal/oblique asymptote
![Page 37: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/37.jpg)
982
x
xxyExample: Sketch the graph of , clearly indicating any
asymptotes and any points where the graph meets the axes.
![Page 38: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/38.jpg)
982
x
xxyExample: Sketch the graph of , clearly indicating any
asymptotes and any points where the graph meets the axes.
• vertical asymptotes at 3x
![Page 39: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/39.jpg)
x
y
x = 3x = - 3
![Page 40: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/40.jpg)
982
x
xxyExample: Sketch the graph of , clearly indicating any
asymptotes and any points where the graph meets the axes.
• vertical asymptotes at 3x
• oblique asymptote at xy
![Page 41: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/41.jpg)
x
y
x = 3x = - 3
y = x
![Page 42: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/42.jpg)
982
x
xxyExample: Sketch the graph of , clearly indicating any
asymptotes and any points where the graph meets the axes.
• vertical asymptotes at 3x
• oblique asymptote at xy • curve meets oblique asymptote at 8x = 0
x = 0
![Page 43: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/43.jpg)
x
y
x = 3x = - 3
y = x
![Page 44: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/44.jpg)
x
y
x = 3x = - 3
y = x
![Page 45: 11X1 T13 03 polynomial division](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559c7e6d8b42a93208b4577/html5/thumbnails/45.jpg)
Exercise 4C; 1c, 2bdfh, 3bd, 4ac, 6ace, 7b, 8, 11, 14*