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*