// Numbas version: exam_results_page_options {"name": "Factorisation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "statement": "

Factorise the following expressions.

Note: Use an asterisk (*) for multiplication and a hat (^) for raising to a power when entering your answers. For example:

$xy$ should be written as $x$*$y$

$a(b+c)$ should be typed as $a$*$(b+c)$

$x^2$ should be typed as $x$^2.

Do check the 'displayed equation' next to where you're typing your answer to check it looks like it should.

Also, the steps provided under 'Show steps' give worked solutions for you to study. Once the concepts are clear, it may be useful to click 'Try another question like this one' to refresh the numbers and complete the question without checking the steps.

", "extensions": [], "tags": [], "advice": "

Follow the steps outlined above to help you factorise the questions.

Click on 'Try another question like this' to try other questions using different numbers.

Alternatively, click on this link to watch a video on factorising expressions.

", "variable_groups": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Factorising polynomials using the highest common factor.

Adapted from 'Factorisation' by Steve Kilgallon.

"}, "parts": [{"prompt": "

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

", "checkvariablenames": true, "answer": "2({c[0]}/2-({c[1]}/2)x^2)", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "vsetrange": [0, 1], "variableReplacements": [], "answersimplification": "all", "scripts": {}, "marks": 1, "musthave": {"message": "

Your answer must be factorised

", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}, "expectedvariablenames": ["x", "a", "b", "c", "p", "q", "l", "m", "y"], "showCorrectAnswer": true, "showpreview": true, "type": "jme", "checkingaccuracy": 0.001, "vsetrangepoints": 5, "stepsPenalty": 0, "checkingtype": "absdiff", "steps": [{"marks": 0, "prompt": "

First, find the highest common factor between the two coefficients. In this case it is $2$. The HCF, therefore, is placed outside of the brackets: $2(\\;\\;\\;\\;)$

Next, put into the brackets the terms which, when multiplied by $2$, result in the the original expression.

To find this, work backwards by dividing each term by the common factor.

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

", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "information", "variableReplacements": [], "scripts": {}}]}, {"prompt": "

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

", "checkvariablenames": true, "answer": "2b*(({c[2]}a/2)+({c[3]}c/2))", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "vsetrange": [0, 1], "variableReplacements": [], "answersimplification": "all", "scripts": {}, "marks": 1, "musthave": {"message": "

Your answer must be factorised

First, find the highest common factor between the two coefficients. In this case it is $2b$. The HCF, therefore, is placed outside of the brackets: $2b(\\;\\;\\;\\;)$

Next, put into the brackets the terms which, when multiplied by two $2b$, result in the the original expression.

To find this, work backwards by dividing each term by the common factor.

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

", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "information", "variableReplacements": [], "scripts": {}}]}, {"prompt": "

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

", "checkvariablenames": true, "answer": "2a*({c[4]}a/2+{c[5]}b/2)", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "vsetrange": [0, 1], "variableReplacements": [], "answersimplification": "all", "scripts": {}, "marks": 1, "musthave": {"message": "

Your answer must be factorised

First, find the highest common factor between the two coefficients. In this case it is $2a$. The HCF, therefore, is placed outside of the brackets: $2a(\\;\\;\\;\\;)$

Next, put into the brackets the terms which, when multiplied by $2a$, result in the the original expression.

To find this, work backwards by dividing each term by the common factor.

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

", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "information", "variableReplacements": [], "scripts": {}}]}, {"prompt": "

$pq^3-p^3q=$

", "checkvariablenames": true, "answer": "p*q*((q^2)-p^2)", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "vsetrange": [0, 1], "variableReplacements": [], "answersimplification": "all", "scripts": {}, "marks": 1, "musthave": {"message": "

Your answer must be factorised

First, find the highest common factor between the two coefficients. In this case it is $pq$. The HCF, therefore, is placed outside of the brackets: $pq(\\;\\;\\;\\;)$

Next, put into the brackets the terms which, when multiplied by $pq$, result in the the original expression.

To find this, work backwards by dividing each term by the common factor.

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

", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "information", "variableReplacements": [], "scripts": {}}]}, {"prompt": "

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

", "checkvariablenames": true, "answer": "3*x*y*(({c2[0]}x/3)+({c2[1]}y^3/3))", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "vsetrange": [0, 1], "variableReplacements": [], "answersimplification": "all", "scripts": {}, "marks": 1, "musthave": {"message": "

Your answer must be factorised

First, find the highest common factor between the two coefficients. In this case it is $3xy$. The HCF, therefore, is placed outside of the brackets: $3xy(\\;\\;\\;\\;)$

Next, put into the brackets the terms which, when multiplied by $3xy$, result in the the original expression.

To find this, work backwards by dividing each term by the common factor.

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

", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "information", "variableReplacements": [], "scripts": {}}]}, {"prompt": "

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

", "checkvariablenames": false, "answer": "({{squares}[0]}+{{squares}[1]}*x)({{squares}[0]}-{{squares}[1]}*x)", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "vsetrange": [0, 1], "variableReplacements": [], "answersimplification": "!basic", "scripts": {}, "marks": 1, "expectedvariablenames": [], "showCorrectAnswer": true, "showpreview": true, "type": "jme", "checkingaccuracy": 0.001, "vsetrangepoints": 5, "stepsPenalty": 0, "checkingtype": "absdiff", "steps": [{"marks": 0, "prompt": "

An expression such this is known as 'the difference of two squares'. This occurs when one square number or term in subtracted from another. The general pattern observed is the following:

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

This because, when the brackets on the right hand side are expanded, the terms simplify:

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

There may be other varables involved, but the same still applies to the term as a whole. For example:

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

Assigning values to coefficients, another example might be:

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

", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "information", "variableReplacements": [], "scripts": {}}]}], "variables": {"firstsquare": {"group": "Ungrouped variables", "templateType": "anything", "definition": "squares[0]^2", "description": "", "name": "firstsquare"}, "squares": {"group": "Ungrouped variables", "templateType": "anything", "definition": "shuffle(2..7)[0..2]", "description": "", "name": "squares"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "shuffle([2,2,6,8,10,14])", "description": "

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

", "name": "c"}, "c3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "shuffle([2,4,6,10,14])[0..3]", "description": "

Part f (HCF:2)

", "name": "c3"}, "q4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except 0)", "description": "", "name": "q4"}, "c2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "shuffle([3,3,6,9,15])[0..5]", "description": "

Coefficients in e,f (HCF: 3)

", "name": "c2"}, "qb": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(q3*q2)+(q1*q4)", "description": "", "name": "qb"}, "q1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "description": "", "name": "q1"}, "qc": {"group": "Ungrouped variables", "templateType": "anything", "definition": "q2*q4", "description": "", "name": "qc"}, "q3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except 0)", "description": "", "name": "q3"}, "secondsquare": {"group": "Ungrouped variables", "templateType": "anything", "definition": "squares[1]^2", "description": "", "name": "secondsquare"}, "qa": {"group": "Ungrouped variables", "templateType": "anything", "definition": "q1*q3", "description": "", "name": "qa"}, "q2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except 0)", "description": "", "name": "q2"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "name": "Factorisation", "preamble": {"js": "", "css": ""}, "rulesets": {}, "ungrouped_variables": ["c", "c2", "c3", "squares", "firstsquare", "secondsquare", "q1", "q2", "q3", "q4", "qa", "qb", "qc"], "type": "question", "contributors": [{"name": "Rachel Staddon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/901/"}]}]}], "contributors": [{"name": "Rachel Staddon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/901/"}]}