MATHHX B
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1.2 Bogstavregning
Gymnasiematematik er mere abstrakt end folkeskolematematik. Her regner vi ofte med bogstaver, så det er vigtigt at få en god forståelse for bogstavregning fra starten af.
Reduktion
At reducere et matematisk udtryk betyder at skrive det på en mere simpel form.
Øvelse 1.2.1
Reducer
-
a) \(b+c-2a+a-c\)
-
b) \(a+a+a\)
-
c) \(b+x-x-b\)
Løsning 1.2.1
-
a) \(b-a\)
-
b) \(3a\)
-
c) \(0\)
Vi husker at \(ab\) betyder \(a\cdot b\), og det kan ikke regnes sammen med \(a\) eller \(b\).
Øvelse 1.2.2
Reducer
-
a) \(ab-a+a-2b+b\)
-
b) \(bc+cb\)
-
c) \(ab+ab+ba\)
Løsning 1.2.2
-
a) \(ab-b\)
-
b) \(2bc\)
-
c) \(3ab\)
Parenteser
En parentes betyder, at man skal regne det, som står i parentesen først. Står der et tal foran (eller bagved) en parentes, betyder det ”gange”. Har vi f.eks. \(2(5+3)\), betyder det altså \(2\cdot (5+3)\).
Øvelse 1.2.3
Regn
-
a) \(2(3-1)\)
-
b) \((5+2)3\)
Løsning 1.2.3
-
a) \(2(3-1)=4\)
-
b) \((5+2)3=21\)
Når der optræder bogstaver i regnestykket, kan man ikke altid regne parentesen først. Så kan man i stedet benytte følgende regler:
-
• Man ganger ind i parenteser ved at gange i hvert led. Hvis du er i tvivl om hvad det betyder, så tjek eksemplet nedenunder.
-
• Plusparenteser kan bare ophæves.
-
• Minusparenteser klares ved at skifte fortegn. Altså man skifter plus til minus og minus til plus.
Øvelse 1.2.4
Reducer
-
a) \(3(x+y)\)
-
b) \((a-b)2\)
-
c) \(2+(x+y)-3\)
-
d) \(-(v+w)\)
-
e) \(a(2b+b)-ba-(2a-2)\)
Faktorisering
Dette afsnit kan springes over, indtil du får brug for det.
At faktorisere betyder at skrive noget som et gangestykke. I praksis svarer det til det modsatte af at gange en parentes ud. Dvs. i stedet for at ophæve en parentes tryller man en parentes frem.
-
Eksempel 1.2.5
Vi vil faktoriserer udtrykket \(2x+2\). Da \(2\) går igen i begge led, kan vi sætte det ud foran en parentes:
\[2x+6=2(x+3)\]
Vi tjekker, at det passer ved at gange parentesen ud:
\[2(x+3)=2x+6\]
Det passede. Udtrykket \(2x+6\) kan altså faktoriseres til \(2(x+3)\).
Løsning 1.2.5
-
a) \(4x+4=4(x+1)\)
-
b) \(3x-15=3(x-5)\)
Man kan også faktorisere med bogstaver.
-
Eksempel 1.2.6
Vi vil faktorisere udtrykket \(3a+7ab\). Vi ser at \(a\) går op i begge led, så vi kan faktorisere med \(a\)
\[3a+7ab=a(3+7b)\]
Tjek selv, at det passer, når man ganger parentesen ud.
Nogle gange er der flere måder, man kan faktorisere et udtryk.
-
Eksempel 1.2.7
Vi vil faktorisere udtrykket \(x^3-x^2\). Vi ser, at \(x\) går op i begge led, så vi kan faktorisere med \(x\)
\[x^3-x^2=x(x^2-x)\]
...men hmm i stedet for at faktorisere med \(x\), kunne vi have været mere aggressive (go big or go home) og faktoriseret med \(x^2\),
\[x^3-x^2=x^2(x-1)\]
Typisk vil man gerne have mest muligt ud foran parentesen.
Øvelse 1.2.6
Faktoriser følgende udtryk mest muligt.
-
a) \(ab+2b\)
-
b) \(x^2-2x\)
-
c) \(3x^4-2x^3+x^2\)
-
d) \(abc+bcd+bcx\)
Det følgende ekstraafsnit forudsætter, at du har regnet afsnit 1.1. Det er et vigtigt afsnit, hvis du gerne vil have en solid forståelse.
Ekstra: Korrekt sprogbrug
Vi starter af med at indføre nogle begreber.
Sum, differens og led
En sum er et udtryk på formen
\[a+b+c+\cdots \]
Her kaldes \(a,b,c\ldots \) for led. En sum består altså af to eller flere led adskilt af et plustegn. En differens er et udtryk på formen
\[a-b\]
Her kaldes \(a\) og \(b\) også for led.
Produkt, faktorer
Et produkt er et udtryk på formen
\[a\cdot b\cdot c\cdots \]
Her kaldes \(a,b,c,\ldots \) for faktorer. Et produkt består altså af to eller flere faktorer adskilt af gangetegn. Vi skriver ofte produkter uden gangetegn. Skriver vi \(ab\) betyder det altså \(a\cdot b\).
Potens, grundtal og eksponent
En potens er et udtryk på formen
\[a^p\]
Her kaldes \(a\) for grundtallet og \(p\) for eksponenten.
-
Eksempel 1.2.8
Betragt udtrykket:
\[a(b+c)\]
Dette betyder \(a\cdot (b+c)\) og derfor er det et produkt bestående af faktorerne \(a\) og \((b+c)\). Den anden faktor, altså \((b+c)\), er en sum bestående af de to led \(b\) og \(c\).
Øvelse 1.2.7
Forklar hvordan udtrykket er opbygget.
-
a) \(2ab\)
-
b) \(a-ab\)
-
c) \(q^p\)
Løsning 1.2.7
-
a) Det er et produkt bestående af faktorerne \(2\), \(a\) og \(b\).
-
b) Det er en differens bestående af ledene \(a\) og \(ab\). Det sidste led er et produkt bestående af faktorerne \(a\) og \(b\).
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c) Det er en potens , hvor grundtallet er \(q\) og eksponenten \(p\).
Ekstra: brøker med bogstaver
Når vi fremover støder på brøker, så vil de ofte være med bogstaver i stedet for tal. Det er dog præcis de samme regler, der gælder.
Forlænge og forkorte
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Eksempel 1.2.9
Vi vil forlænge brøken \(\frac {a+b}{c}\) med \(5\).
\[\frac {a+b}{c}=\frac {5(a+b)}{5c}=\frac {5a+5b}{5c}\]
Læg mærke til parentesen efter første lighedstegn. Det er en almindelig fejl, at man glemmer den. Glemmer man den, får man ikke ganget hele tælleren med \(5\), men kun første del.
Hvis der optræder flere led (vi husker at led er adskilt af plus og minus) i tæller eller nævner, skal man forkorte i hvert led.
Øvelse 1.2.8
Regn
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a) Forlæng brøken \(\frac {a+1}{b}\) med \(c\).
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b) Forkort brøken \(\frac {ab}{bc}\).
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c) Forkort brøken \(\frac {ab}{b}\)
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d) Forkort brøken \(\frac {ab+ac-a^3}{2a^2+a}\)
Plus og minus
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Eksempel 1.2.12
Vi vil regne \(\frac {2a+b}{a}+\frac {4}{b}\). Vi forlænger første brøk med \(b\) og anden brøk med \(a\):
\(\seteqnumber{0}{1.}{0}\)
\begin{align*}
\frac {2a+b}{a}+\frac {4}{b}&=\frac {(2a+b)b}{ab}+\frac {4a}{ab}\\ &=\frac {(2a+b)b+4a}{ab}\\ &=\frac {2ab+b^2+4a}{ab}
\end{align*}
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Eksempel 1.2.13
Vi vil regne \(\frac {a}{b}+c\)
\(\seteqnumber{0}{1.}{0}\)
\begin{align*}
\frac {a}{b}+c & = \frac {a}{b}+\frac {c}{1}\\ &=\frac {a}{b}+\frac {c\cdot b}{1\cdot b}\\ &=\frac {a}{b}+\frac {bc}{b}\\ &=\frac {a+bc}{b}
\end{align*}
Øvelse 1.2.9
Regn
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a) \(\frac {a}{b}+\frac {c}{b}\)
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b) \(\frac {a}{b}+\frac {b}{a}\)
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c) \(\frac {a}{b}-2\)
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d) \(\frac {b^2}{4a^2}-\frac {c}{a}\)
Løsning 1.2.9
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a) \(\frac {a}{b}+\frac {c}{b}=\frac {a+c}{b}\)
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b) \(\frac {a}{b}+\frac {b}{a}=\frac {a^2+b^2}{ab}\)
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c) \(\frac {a}{b}-2=\frac {a-2b}{b}\)
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d) \(\frac {b^2}{4a^2}-\frac {c}{a}=\frac {b^2-4ac}{4a^2}\)
Gange og dividere
Når man ganger eller dividerer brøker med flere led i tæller/nævner, skal man huske at sætte parenteser.
Øvelse 1.2.10
Regn:
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a) \(\frac {a+b}{c}\cdot \frac {a}{b}\)
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b) \(\frac {a+2}{b}\cdot a\)
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c) \(\frac {\ \frac {2}{a}\ }{b}\)
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d) \(\frac {\ a\ }{\frac {2}{a}}\)
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e) \(\frac {\ \frac {3+a}{b}\ }{\frac {a}{b}}\)
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f) \(\frac {1}{\frac {1}{y}}\)
Løsning 1.2.10
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a) \(\frac {a+b}{c}\cdot \frac {a}{b}=\frac {a^2+ab}{bc}\)
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b) \(\frac {a+2}{b}\cdot a = \frac {a^2+2a}{b}\)
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c) \(\frac {\ \frac {2}{a}\ }{b}=\frac {2}{ab}\)
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d) \(\frac {\ a\ }{\frac {1}{a}}=a^2\)
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e) \(\frac {\ \frac {3+a}{b}\ }{\frac {a}{b}}= \frac {3+a}{a}\)
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f) \(\frac {1}{\frac {1}{y}}=y\)