11X1 T16 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 xxxxx2x

  • 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 xxxxx2x

    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 xxxxx2x

    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 xxxxx2x

    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 xxxxx2x

    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 xxxxx2x

    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 xxxxx2x

    234 0 xxx 2

    22x3

    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 xxxxx2x

    234 0 xxx 2

    22x3

    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 xxxxxx22x

  • 1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf

    (i) Divide the polynomial

    321210213 2342 xxxxxx22x

    234 262 xxx

  • 1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf

    (i) Divide the polynomial

    321210213 2342 xxxxxx22x

    234 262 xxx 23 104 xx

  • 1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf

    (i) Divide the polynomial

    321210213 2342 xxxxxx22x

    234 262 xxx 23 104 xx 32 x

  • 1988 Extension 1 HSC Q4b) 13by 3212102 2234 xxxg xxxxxf

    (i) Divide the polynomial

    321210213 2342 xxxxxx22x

    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 xxxxxx22x

    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 xxxxxx22x

    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 xxxxxx22x

    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 xxxxxx22x

    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 xxxxxx22x

    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 xxxxxx22x

    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 xxxxxx22x

    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 xxxxxx22x

    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*

    Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Slide Number 39Slide Number 40Slide Number 41Slide Number 42Slide Number 43Slide Number 44Slide Number 45