11x1 t13 03 polynomial division
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Polynomial Division
Polynomial Division xRxQxAxP
where;
Polynomial Division xRxQxAxP
where;A(x) is the divisor
Polynomial Division xRxQxAxP
where;A(x) is the divisor Q(x) is the quotient
Polynomial Division xRxQxAxP
where;A(x) is the divisor Q(x) is the quotientR(x) is the remainder
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:
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
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
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
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
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
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
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
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
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
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
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
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
321210213 2342 xxxxxx
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
321210213 2342 xxxxxx
22x
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
321210213 2342 xxxxxx
22x
234 262 xxx
1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf
(i) Divide the polynomial
321210213 2342 xxxxxx
22x
234 262 xxx 23 104 xx
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
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
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
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
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
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
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
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
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.
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
Further graphing of polynomials
xAxRxQ
xAxPy
Further graphing of polynomials
xAxRxQ
xAxPy
solve A(x) = 0 to find vertical asymptotes
Further graphing of polynomials
xAxRxQ
xAxPy
solve A(x) = 0 to find vertical asymptotes
y = Q(x) is the horizontal/oblique
asymptote
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
982
x
xxyExample: Sketch the graph of , clearly indicating any
asymptotes and any points where the graph meets the axes.
982
x
xxyExample: Sketch the graph of , clearly indicating any
asymptotes and any points where the graph meets the axes.
• vertical asymptotes at 3x
x
y
x = 3x = - 3
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
x
y
x = 3x = - 3
y = x
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
x
y
x = 3x = - 3
y = x
x
y
x = 3x = - 3
y = x
Exercise 4C; 1c, 2bdfh, 3bd, 4ac, 6ace, 7b, 8, 11, 14*
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