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Factorising polynomials using the highest common factor.

Adapted from 'Factorisation' by Steve Kilgallon.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Faktoriser følgende uttrykk:

Merk: Bruk stjerne (*) til multiplikasjon og hatt (^) for å opphøye i en potens. For eksempel, $xy$ skrives som $x$*$y$, $a(b+c)$ skrives som $a$*$(b+c)$ og $x^2$ som $x$^2.

Under \"Vis tips\" og \"Vis svar\" finnes fullstendig løsning til oppgavene.

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Coefficients in a,b,c (HCF: 2)

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Part f (HCF:2)

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Coefficients in e,f (HCF: 3)

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$\\var{c[0]}-\\var{c[1]}x^2=$

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Først finner vi største felles faktor, i dette tilfelle $2$. Vi setter den så utenfor parentes: $2(\\;\\;\\;\\;)$

Deretter må vi finne ut hva som står igjen i leddene inne i parentesen.

For å finne dette kan vi dele hvert ledd med den felles faktoren:

$2\\big(\\frac{\\var{c[0]}}{2}-\\frac{\\var{c[1]}}{2}x^2\\big)=2\\big(\\simplify{{c[0]}/2}-\\simplify{{c[1]}/2}x^2\\big)$

"}], "answer": "2({c[0]}/2-({c[1]}/2)x^2)", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "

Your answer must be factorised

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$\\var{c[2]}ab+\\var{c[3]}bc=$

Først finner vi største felles faktor, i dette tilfelle $2b$. Vi setter den så utenfor parentes: $2b(\\;\\;\\;\\;)$

Deretter må vi finne ut hva som står igjen i leddene inne i parentesen.

For å finne dette kan vi dele hvert ledd med den felles faktoren:

$2b\\big(\\frac{\\var{c[2]}ab}{2b}+\\frac{\\var{c[3]}bc}{2b}\\big)=2b\\big(\\simplify{({c[2]}/2)*a}+\\simplify{({c[3]}/2)*c}\\big)$

"}], "answer": "2b*(({c[2]}a/2)+({c[3]}c/2))", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "

Your answer must be factorised

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$\\var{c[4]}a^2+\\var{c[5]}ab=$

Først finner vi største felles faktor, i dette tilfelle $2a$. Vi setter den så utenfor parentes: $2a(\\;\\;\\;\\;)$

Deretter må vi finne ut hva som står igjen i leddene inne i parentesen.

For å finne dette kan vi dele hvert ledd med den felles faktoren:

$2a\\big(\\frac{\\var{c[4]}a^2}{2a}+\\frac{\\var{c[5]}ab}{2a}\\big)=2a\\big(\\simplify{({c[4]}/2)*a}+\\simplify{({c[5]}/2)*b}\\big)$

"}], "answer": "2a*({c[4]}a/2+{c[5]}b/2)", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "

Your answer must be factorised

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$pq^3-p^3q=$

Først finner vi største felles faktor, i dette tilfelle $pq$. Vi setter den så utenfor parentes: $pq(\\;\\;\\;\\;)$

Deretter må vi finne ut hva som står igjen i leddene inne i parentesen.

For å finne dette kan vi dele hvert ledd med den felles faktoren:

$pq\\big(\\frac{pq^3}{pq}-\\frac{p^3q}{pq}\\big)=pq(q^2-p^2)$

"}], "answer": "p*q*((q^2)-p^2)", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "

Your answer must be factorised

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$\\var{c2[0]}x^2y+\\var{c2[1]}xy^4=$

Først finner vi største felles faktor, i dette tilfelle $3xy$. Vi setter den så utenfor parentes: $3xy(\\;\\;\\;\\;)$

Deretter må vi finne ut hva som står igjen i leddene inne i parentesen.

For å finne dette kan vi dele hvert ledd med den felles faktoren:

$3xy\\big(\\frac{\\var{c2[0]}x^2y}{3xy}+\\frac{\\var{c2[1]}xy^4}{3xy}\\big)=3xy\\big(\\simplify{({c2[0]}/3)*x}+\\simplify{({{c2}[1]}/3)*y^3}\\big)$

"}], "answer": "3*x*y*(({c2[0]}x/3)+({c2[1]}y^3/3))", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "

Your answer must be factorised

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$\\var{c3[0]}p^3q-\\var{c3[1]}p^2q^2+\\var{c3[2]}pq^3=$

Først finner vi største felles faktor, i dette tilfelle $2pq$. Vi setter den så utenfor parentes: $2pq(\\;\\;\\;\\;)$

Deretter må vi finne ut hva som står igjen i leddene inne i parentesen.

For å finne dette kan vi dele hvert ledd med den felles faktoren:

$2pq\\big(\\frac{\\var{c3[0]}p^3q}{2pq}-\\frac{\\var{c3[1]}p^2q^2}{2pq}+\\frac{\\var{c3[2]}pq^3}{2pq}\\big)=2pq\\big(\\simplify{({c3[0]}/2)p^2}-\\simplify{({c3[1]}/2)*p q}+\\simplify{({c3[2]}/2)*q^2}\\big)$

"}], "answer": "2*p*q*({c3[0]}p^2/2-{c3[1]}*p*q/2+{c3[2]}q^2/2)", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "

Your answer must be factorised

$\\var{c2[2]}lm^2-\\var{c2[3]}l^3m^3+\\var{c2[4]}l^2m^4=$

Først finner vi største felles faktor, i dette tilfelle $3lm^2$. Vi setter den så utenfor parentes: $3lm^2(\\;\\;\\;\\;)$

Deretter må vi finne ut hva som står igjen i leddene inne i parentesen.

For å finne dette kan vi dele hvert ledd med den felles faktoren:

$3lm^2\\big(\\frac{\\var{c2[2]}lm^2}{3lm^2}-\\frac{\\var{c2[3]}l^3m^3}{3lm^2}+\\frac{\\var{c2[4]}l^2m^4}{3lm^2}\\big)=3lm^2\\big(\\simplify{({c2[2]}/3)}-\\simplify{({c2[3]}/3)*l^2m}+\\simplify{({c2[4]}/3)*lm^2}\\big)$

"}], "answer": "3*l*m^2*({c2[2]}/3-{c2[3]}*l^2*m/3+{c2[4]}l*m^2/3)", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "

Your answer must be factorised

"}, "valuegenerators": [{"name": "l", "value": ""}, {"name": "m", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

$\\var{firstsquare}-\\var{secondsquare}x^2$ =

Dette kjenner vi som 3.kvadratsetning, eller konjugatsetningen:

$a^2-b^2=(a+b)(a-b)$

Vi kan multiplisere parentesene på høyre side for å vise at det stemmer::

$(a+b)(a-b)=a^2-ab+ab-b^2=a^2-b^2$

$ \"a\" $ og $ \"b\" $ kan være byttet ut med mer sammensatte uttrykk, husk da at hver faktor i leddene skal opphøyes i andre:

$a^2x^4-b^2y^6=(ax^2+by^3)(ax^2-by^3)$

Leddene kan også være kombinasjoner av tall og bokstaver:

$9x^2-16y^4=(3x+4y^2)(3x-4y^2)$

"}], "answer": "({{squares}[0]}+{{squares}[1]}*x)({{squares}[0]}-{{squares}[1]}*x)", "answerSimplification": "!basic", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Torris Bakke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/942/"}], "resources": []}]}], "contributors": [{"name": "Torris Bakke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/942/"}]}