# Oppgaver: Polynomdivisjon
:::::::::::::::{exercise} Oppgave 1
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level: 1
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::::::::::::::{tab-set}
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class: tabs-parts
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:::::::::::::{tab-item} a
Regn ut
$$
(x^3 - 5x^2 - 9x + 45) : (x - 5)
$$
:::{answer}
$$
(x^3 - 5x^2 - 9x + 45) : (x - 5) = x^2 - 9.
$$
:::
::::{solution}
:::{polydiv}
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p: x^3 - 5x^2 - 9x + 45
q: x - 5
width: 70%
---
:::
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:::::::::::::{tab-item} b
Regn ut
$$
(x^3 - 2x^2 - 11x + 12) : (x - 4)
$$
:::{answer}
$$
(x^3 - 2x^2 - 11x + 12) : (x - 4) = x^2 + 2x - 3.
$$
:::
::::{solution}
:::{polydiv}
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p: x^3 - 2x^2 - 11x + 12
q: x - 4
width: 80%
---
:::
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:::::::::::::{tab-item} c
Regn ut
$$
(x^3 + 11x^2 + 38x + 40) : (x + 5)
$$
:::{answer}
$$
(x^3 + 11x^2 + 38x + 40) : (x + 5) = x^2 + 6x + 8.
$$
:::
::::{solution}
:::{polydiv}
---
p: x^3 + 11x^2 + 38x + 40
q: x + 5
width: 80%
---
:::
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:::::::::::::{tab-item} d
Regn ut
$$
(x^3 + 3x^2 - 4x - 12) : (x + 2)
$$
:::{answer}
$$
(x^3 + 3x^2 - 4x - 12) : (x + 2) = x^2 + x - 6.
$$
:::
::::{solution}
:::{polydiv}
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p: x^3 + 3x^2 - 4x - 12
q: x + 2
width: 80%
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:::::::::::::::{exercise} Oppgave 2
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level: 1
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::::::::::::::{tab-set}
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class: tabs-parts
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:::::::::::::{tab-item} a
Regn ut
$$
(x^3 - 3x^2 - 24x + 80) : (x + 4)
$$
:::{answer}
$$
(x^3 - 3x^2 - 24x + 80) : (x + 4) = x^2 - 7x + 4 + \dfrac{64}{x + 4}
$$
:::
::::{solution}
:::{polydiv}
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p: x^3 - 3x^2 - 24x + 80
q: x + 4
width: 80%
---
:::
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:::::::::::::{tab-item} b
Regn ut
$$
(x^3 + 2x^2 - 16x - 32) : (x - 3)
$$
:::{answer}
$$
(x^3 + 2x^2 - 16x - 32) : (x - 3) = x^2 + 5x - 1 + \dfrac{-35}{x - 3}
$$
:::
::::{solution}
:::{polydiv}
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p: x^3 + 2x^2 - 16x - 32
q: x - 3
width: 80%
---
:::
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:::::::::::::{tab-item} c
Regn ut
$$
(x^3 - 4x^2 - 5x) : (x + 4)
$$
:::{answer}
$$
(x^3 - 4x^2 - 5x) : (x + 4) = x^2 - 8x + 27 + \dfrac{-108}{x + 4}
$$
:::
::::{solution}
:::{polydiv}
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p: x^3 - 4x^2 - 5x
q: x + 4
width: 80%
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:::
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:::::::::::::{tab-item} d
Regn ut
$$
(x^3 - 9x) : (x - 5)
$$
:::{answer}
$$
(x^3 - 9x) : (x - 5) = x^2 + 5x + 16 + \dfrac{80}{x - 5}
$$
:::
::::{solution}
:::{polydiv}
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p: x^3 - 9x
q: x - 5
width: 80%
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:::::::::::::::{exercise} Oppgave 3
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level: 1
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Utfør polynomdivisjonene
::::::::::::::{tab-set}
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class: tabs-parts
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:::::::::::::{tab-item} a
$$
(x^3 + x^2 - 9x - 9) : (x + 3)
$$
:::::{answer}
$$
(x^3 + x^2 - 9x - 9) : (x + 3) = x^2 - 2x - 3
$$
:::::
::::{solution}
:::{polydiv}
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p: x^3 + x^2 - 9x - 9
q: x + 3
width: 80%
---
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:::::::::::::{tab-item} b
$$
(x^3 - 4x^2 - 11x + 30) : (x - 2)
$$
:::::{answer}
$$
(x^3 - 4x^2 - 11x + 30) : (x - 2) = x^2 - 2x - 15
$$
:::::
::::{solution}
:::{polydiv}
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p: x^3 - 4x^2 - 11x + 30
q: x - 2
width: 80%
---
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:::::::::::::{tab-item} c
$$
(x^4 - x^3 - 7x^2 + x + 6) : (x + 2)
$$
:::::{answer}
$$
(x^4 - x^3 - 7x^2 + x + 6) : (x + 2) = x^3 - 3x^2 - x + 3
$$
:::::
::::{solution}
:::{polydiv}
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p: x^4 - x^3 - 7x^2 + x + 6
q: x + 2
width: 80%
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:::::::::::::{tab-item} d
$$
(x^3 - 2x^2 - 5x + 6) : (x - 1)
$$
:::::{answer}
$$
(x^3 - 2x^2 - 5x + 6) : (x - 1) = x^2 - x - 6
$$
:::::
::::{solution}
:::{polydiv}
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p: x^3 - 2x^2 - 5x + 6
q: x - 1
width: 80%
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:::::::::::::::{exercise} Oppgave 4
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level: 1
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Utfør polynomdivisjonene
::::::::::::::{tab-set}
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class: tabs-parts
---
:::::::::::::{tab-item} a
$$
(x^3 - 5x^2 - 2x + 6) : (x + 1)
$$
:::::{answer}
$$
(x^3 - 5x^2 - 2x + 6) : (x + 1) = x^3 - 6x + 4 + \dfrac{2}{x + 1}
$$
:::::
::::{solution}
:::{polydiv}
---
p: x^3 - 5x^2 - 2x + 6
q: x + 1
width: 80%
---
:::
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:::::::::::::{tab-item} b
$$
(x^3 + x^2 - 5x + 3) : (x^2 + 2x - 3)
$$
:::::{answer}
$$
(x^3 + x^2 - 5x + 3) : (x^2 + 2x - 3) = x - 1
$$
:::::
::::{solution}
:::{polydiv}
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p: x^3 + x^2 - 5x + 3
q: x^2 + 2x - 3
width: 80%
---
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:::::::::::::{tab-item} c
$$
(x^3 + 6x^2 - x - 30) : (x - 1)
$$
:::::{answer}
$$
(x^3 + 6x^2 - x - 30) : (x - 1) = x^2 + 7x + 6 + \dfrac{-24}{x - 1}
$$
:::::
::::{solution}
:::{polydiv}
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p: x^3 + 6x^2 - x - 30
q: x - 1
width: 80%
---
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:::::::::::::{tab-item} d
$$
(x^4 - 10x^3 + 35x^2 - 50x + 24) : (x^2 - 6x + 8)
$$
:::::{answer}
$$
(x^4 - 10x^3 + 35x^2 - 50x + 24) : (x^2 - 6x + 8) = x^2 - 4x + 3
$$
:::::
::::{solution}
:::{polydiv}
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p: x^4 - 10x^3 + 35x^2 - 50x + 24
q: x^2 - 6x + 8
width: 100%
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:::{margin} Tips: Oppgave 5
Hint: Hvis det "mangler" noen ledd i dividenden, kan det være lurt å legge til en null på plassen til leddet som mangler. For eksempel
$$
x^3 - x + 1 = x^3 + 0x^2 - x + 1
$$
:::
:::::::::::::::{exercise} Oppgave 5
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level: 2
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Utfør polynomdivisjonene.
::::::::::::::{tab-set}
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class: tabs-parts
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:::::::::::::{tab-item} a
$$
(x^3 + 2x - 1) : (x + 2)
$$
:::::{answer}
$$
(x^3 + 2x - 1) : (x + 2) = x^2 - 2x + 6 + \dfrac{-13}{x + 2}
$$
:::::
::::{solution}
:::{polydiv}
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p: x^3 + 2x - 1
q: x + 2
width: 80%
---
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:::::::::::::{tab-item} b
$$
(x^3 - 7x + 1) : (x^2 - 3)
$$
:::::{answer}
$$
(x^3 - 7x + 1) : (x^2 - 3) = x + \dfrac{-4x + 1}{x^2 - 3}
$$
:::::
::::{solution}
:::{polydiv}
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p: x^3 - 7x + 1
q: x^2 - 3
width: 80%
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:::::::::::::{tab-item} c
$$
(4x^3 - 18x^2 + 26x - 15) : (2x^2 - 4x + 3)
$$
:::::{answer}
$$
(4x^3 - 18x^2 + 26x - 15) : (2x^2 - 4x + 3) = 2x - 5
$$
:::::
::::{solution}
:::{polydiv}
---
p: 4x^3 - 18x^2 + 26x - 15
q: 2x^2 - 4x + 3
width: 80%
---
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:::::::::::::{tab-item} d
$$
(x^4 - 1) : (x - 1)
$$
:::::{answer}
$$
(x^4 - 1) : (x - 1) = x^3 + x^2 + x + 1
$$
:::::
::::{solution}
:::{polydiv}
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p: x^4 - 1
q: x - 1
width: 80%
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:::::::::::::::{exercise} Oppgave 6
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level: 2
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::::::::::::::{tab-set}
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class: tabs-parts
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:::::::::::::{tab-item} a
Bestem $a$, $b$ og $c$ slik at likningen blir en identitet.
$$
x^3 - x^2 + 4x - 4 = (x - 1)(ax^2 + bx + c).
$$
:::::{admonition} Fasit
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class: dropdown, answer
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$$
a = 1 \and b = 0 \and c = 4
$$
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:::::{admonition} Løsning
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class: dropdown, solution
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Vi utfører polynomdivisjon med $(x - 1)$ for å finne andregradspolynomet $(ax^2 + bx + c)$
:::{polydiv}
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p: x^3 - x^2 + 4x - 4
q: x - 1
width: 80%
---
:::
Vi ser at
$$
ax^2 + bx + c = x^2 + 4
$$
som betyr at
$$
a = 1 \and b = 0 \and c = 4.
$$
:::::
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:::::::::::::{tab-item} b
Bestem $a$ og $b$ slik at likningen blir en identitet.
$$
3x^3 + 2x^2 - 12x - 8 = (x^2 - 4)(ax + b).
$$
:::::{admonition} Fasit
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class: dropdown, answer
---
$$
a = 3 \and b = 2.
$$
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:::::{admonition} Løsning
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class: dropdown, solution
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Vi utfører polynomdivisjon med $(x^2 - 4)$ for å finne førstegradspolynomet $(ax + b)$.
:::{polydiv}
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p: 3x^3 + 2x^2 - 12x - 8
q: x^2 - 4
width: 80%
---
:::
Vi ser at
$$
ax + b = 3x + 2
$$
som betyr at
$$
a = 3 \and b = 2.
$$
:::::
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:::::::::::::{tab-item} c
Bestem $a$, $b$ og $c$ slik at likningen blir en identitet.
$$
-x^4 + 2x^3 + x^2 - 4x + 2 = (x^2 - 2x + 1)(ax^2 + bx + c).
$$
:::::{admonition} Fasit
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class: dropdown, answer
---
$$
a = -1 \and b = 0 \and c = 2.
$$
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:::::{admonition} Løsning
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class: dropdown, solution
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Vi utfører polynomdivisjon med $(x^2 - 2x + 1)$ for å finne det ukjente andregradspolynomet:
:::{polydiv}
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p: -x^4 + 2x^3 + x^2 - 4x + 2
q: x^2 - 2x + 1
width: 80%
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:::
Vi ser at
$$
ax^2 + bx + c = -x^2 + 2
$$
som betyr at
$$
a = -1 \and b = 0 \and c = 2.
$$
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:::::::::::::::{exercise} Oppgave 7
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level: 2
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En tredjegradsfunksjon er gitt ved
$$
f(x) = x^3 - 4x^2 + 3x - 2.
$$
::::::::::::::{tab-set}
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class: tabs-parts
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:::::::::::::{tab-item} a
Bruk et Horner-skjema til å regne ut $f(2)$.
::::{admonition} Fasit
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class: answer, dropdown
---
$$
f(2) = -4.
$$
**Horner-skjema**:
:::{horner}
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p: x^3 - 4x^2 + 3x - 2
x: 2
width: 60%
---
:::
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::::{solution}
:::{horner}
---
p: x^3 - 4x^2 + 3x - 2
x: 2
width: 60%
tutor:
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:::
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:::::::::::::{tab-item} b
Bruk et Horner-skjema til å regne ut $f(-1)$.
::::{answer}
$$
f(-1) = -10.
$$
**Horner-skjema**:
:::{horner}
---
p: x^3 - 4x^2 + 3x - 2
x: -1
width: 60%
---
:::
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::::{solution}
:::{horner}
---
p: x^3 - 4x^2 + 3x - 2
x: -1
width: 60%
tutor:
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:::
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:::::::::::::{tab-item} c
Bruk et Horner-skjema til å regne ut $f(1)$.
::::{admonition} Fasit
---
class: answer, dropdown
---
$$
f(1) = -2.
$$
**Horner-skjema**:
:::{horner}
---
p: x^3 - 4x^2 + 3x - 2
x: 1
width: 60%
---
:::
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::::{solution}
:::{horner}
---
p: x^3 - 4x^2 + 3x - 2
x: -3
width: 60%
tutor:
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:::::::::::::{tab-item} d
Bruk et Horner-skjema til å regne ut $f(-3)$.
::::{admonition} Fasit
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class: answer, dropdown
---
$$
f(-3) = -74.
$$
**Horner-skjema**:
:::{horner}
---
p: x^3 - 4x^2 + 3x - 2
x: -3
width: 60%
---
:::
::::
::::{solution}
:::{horner}
---
p: x^3 - 4x^2 + 3x - 2
x: -3
width: 60%
tutor:
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:::::::::::::::{exercise} Oppgave 8
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level: 2
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::::::::::::::{tab-set}
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class: tabs-parts
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:::::::::::::{tab-item} a
Bruk et Horner-skjema til å regne ut polynomdivisjonen
$$
(x^3 + x^2 - 5x + 3) : (x + 3).
$$
::::{answer}
$$
(x^3 + x^2 - 5x + 3) : (x + 3) = x^2 - 2x + 1.
$$
:::{horner}
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p: x^3 + x^2 - 5x + 3
x: -3
width: 60%
---
:::
::::
::::{solution}
$$
(x^3 + x^2 - 5x + 3) : (x + 3) = x^2 - 2x + 1.
$$
:::{horner}
---
p: x^3 + x^2 - 5x + 3
x: -3
width: 60%
tutor:
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:::
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:::::::::::::{tab-item} b
Bruk et Horner-skjema til å regne ut polynomdivisjonen
$$
(x^3 - 2x^2 + 1) : (x - 2).
$$
::::{answer}
:::{horner}
---
p: x^3 - 2x^2 + 1
x: 2
width: 60%
---
:::
$$
(x^3 - 2x^2 + 1) : (x - 2) = x^2 + \dfrac{1}{x - 2}
$$
::::
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:::::::::::::{tab-item} c
Bruk et Horner-skjema til å regne ut polynomdivisjonen
$$
(x^4 + 3x^3 - 15x^2 - 19x + 30) : (x - 1).
$$
::::{answer}
:::{horner}
---
p: x^4 + 3x^2 - 15x^2 - 19x + 30
x: 1
width: 60%
---
:::
$$
(x^4 + 3x^3 - 15x^2 - 19x + 30) : (x - 1) = x^3 + x^2 - 11x - 30
$$
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:::::::::::::{tab-item} d
Bruk et Horner-skjema til å regne ut polynomdivisjonen
$$
(x^3 - 8) : (x - 2).
$$
::::{answer}
:::{horner}
---
p: x^3 - 8
x: 2
width: 60%
---
:::
$$
(x^3 - 8) : (x - 2) = x^2 + 2x + 4
$$
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