// Numbas version: finer_feedback_settings {"name": "Terry's copy of Expanding a binomial product (non-monic factors)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"type": "question", "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "pickQuestions": 0, "name": ""}], "variable_groups": [], "variables": {"a": {"group": "Ungrouped variables", "name": "a", "definition": "shuffle(-12..12 except 0)[0..2]", "templateType": "anything", "description": ""}, "b": {"group": "Ungrouped variables", "name": "b", "definition": "shuffle(-12..12 except 0)[0..2]", "templateType": "anything", "description": ""}, "d": {"group": "Ungrouped variables", "name": "d", "definition": "shuffle(-12..12 except 0)[0..2]", "templateType": "anything", "description": ""}, "c": {"group": "Ungrouped variables", "name": "c", "definition": "shuffle(-12..12 except 0)[0..2]", "templateType": "anything", "description": ""}}, "showQuestionGroupNames": false, "preamble": {"js": "", "css": ""}, "tags": ["binomial", "binomial product", "distributive law", "expanding", "factorisation", "Factorisation", "factors", "Factors", "non-monic", "quadratic"], "name": "Terry's copy of Expanding a binomial product (non-monic factors)", "rulesets": {}, "statement": "

Expand and simplify the following.

", "ungrouped_variables": ["a", "b", "c", "d"], "functions": {}, "parts": [{"gaps": [{"vsetrangepoints": 5, "answer": "{c[0]*d[0]}x^2+{d[0]*a[0]+c[0]*b[0]}x+{a[0]*b[0]}", "variableReplacements": [], "vsetrange": [0, 1], "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showpreview": true, "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme", "expectedvariablenames": ["x"], "marks": 1, "scripts": {}, "variableReplacementStrategy": "originalfirst", "notallowed": {"message": "

Ensure you don't use brackets in your answer.

", "strings": ["(", ")"], "showStrings": false, "partialCredit": 0}}], "type": "gapfill", "steps": [{"type": "information", "variableReplacements": [], "marks": 0, "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "prompt": "

Method 1 (the distributive law)

We expand $\\simplify[basic]{({c[0]}x+{a[0]})({d[0]}x+{b[0]})}$ one bracket at a time.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$\\simplify[basic]{({c[0]}x+{a[0]})({d[0]}x+{b[0]})}$	$=$	\n $\\simplify[basic]{{c[0]}x({d[0]}x+{b[0]})+{a[0]}({d[0]}x+{b[0]})}$ \n	\n (each term in one bracket times the entire other bracket) \n
	$=$	$\\simplify[basic]{{c[0]d[0]}x^2+{c[0]b[0]}x+{d[0]a[0]}x+{a[0]b[0]}}$	(use the distributive law on each bracket)
	$=$	$\\simplify[basic]{{c[0]d[0]}x^2+{d[0]a[0]+c[0]b[0]}x+{a[0]b[0]}}$	(collect like terms)

Method 2 (FOIL)

Multiply the First terms in each bracket, then the Outer terms, then the Inner terms and then the Last terms. Add them all together.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$\\simplify[basic]{({c[0]}x+{a[0]})({d[0]}x+{b[0]})}$	$=$	\n $\\simplify[basic]{{c[0]d[0]}x^2+{c[0]b[0]}x+{d[0]a[0]}x+{a[0]b[0]}}$ \n	\n (First, Outer, Inner, Last) \n
	$=$	$\\simplify[basic]{{c[0]d[0]}x^2+{d[0]a[0]+c[0]b[0]}x+{a[0]b[0]}}$	(collect like terms)

"}], "variableReplacements": [], "stepsPenalty": "1", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "variableReplacementStrategy": "originalfirst", "prompt": "

$\\simplify[basic]{({c[0]}x+{a[0]})({d[0]}x+{b[0]})}$ = [[0]]

"}, {"gaps": [{"vsetrangepoints": 5, "answer": "{c[1]*d[1]}x^2+{d[1]*a[1]+c[1]*b[1]}x+{a[1]*b[1]}", "variableReplacements": [], "vsetrange": [0, 1], "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showpreview": true, "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme", "expectedvariablenames": ["x"], "marks": 1, "scripts": {}, "variableReplacementStrategy": "originalfirst", "notallowed": {"message": "

Ensure you don't use brackets in your answer.

Method 1 (the distributive law)

We expand $\\simplify[basic]{({c[1]}x+{a[1]})({d[1]}x+{b[1]})}$ one bracket at a time.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$\\simplify[basic]{({c[1]}x+{a[1]})({d[1]}x+{b[1]})}$	$=$	\n $\\simplify[basic]{{c[1]}x({d[1]}x+{b[1]})+{a[1]}({d[1]}x+{b[1]})}$ \n	\n (each term in one bracket times the entire other bracket) \n
	$=$	$\\simplify[basic]{{c[1]d[1]}x^2+{c[1]b[1]}x+{d[1]a[1]}x+{a[1]b[1]}}$	(use the distributive law on each bracket)
	$=$	$\\simplify[basic]{{c[1]d[1]}x^2+{d[1]a[1]+c[1]b[1]}x+{a[1]b[1]}}$	(collect like terms)

Method 2 (FOIL)

Multiply the First terms in each bracket, then the Outer terms, then the Inner terms and then the Last terms. Add them all together.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$\\simplify[basic]{({c[1]}x+{a[1]})({d[1]}x+{b[1]})}$	$=$	\n $\\simplify[basic]{{c[1]d[1]}x^2+{c[1]b[1]}x+{d[1]a[1]}x+{a[1]b[1]}}$ \n	\n (First, Outer, Inner, Last) \n
	$=$	$\\simplify[basic]{{c[1]d[1]}x^2+{d[1]a[1]+c[1]b[1]}x+{a[1]b[1]}}$	(collect like terms)

"}], "variableReplacements": [], "stepsPenalty": "1", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "variableReplacementStrategy": "originalfirst", "prompt": "

$\\simplify[basic]{({c[1]}x+{a[1]})({d[1]}x+{b[1]})}$ = [[0]]

"}], "variablesTest": {"maxRuns": "127", "condition": ""}, "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "notes": ""}, "advice": "", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Terry Young", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3130/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Terry Young", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3130/"}]}