// Numbas version: exam_results_page_options {"name": "Expanding a binomial product (difference of two squares, perfect squares)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a"], "name": "Expanding a binomial product (difference of two squares, perfect squares)", "tags": ["binomial", "binomial product", "difference of two squares", "distributive law", "expanding", "factorisation", "Factorisation", "Factors", "factors", "monic", "perfect square", "quadratic"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": "1", "prompt": "

$\\simplify{(x+{a[0]})(x-{a[0]})}$ = [[0]]

\n

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "

Ensure you don't use brackets in your answer.

", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "x^2-{a[0]*a[0]}", "marks": 1, "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme"}], "steps": [{"prompt": "

Method 1 (the distributive law)

\n

We expand $\\simplify{(x+{a[0]})(x-{a[0]})}$ 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{(x+{a[0]})(x-{a[0]})}$$=$\n

$\\simplify{x(x-{a[0]})+{a[0]}(x-{a[0]})}$

\n		\n

          (each term in one bracket times the entire other bracket)

\n
$=$$\\simplify{x^2-{a[0]}x+{a[0]}x-{a[0]*a[0]}}$          (use the distributive law on each bracket)
$=$$\\simplify{x^2-{a[0]*a[0]}}$          (collect like terms)
\n

Method 2 (FOIL)

\n

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{(x+{a[0]})(x-{a[0]})}$$=$\n

$\\simplify[basic]{x^2-{a[0]}x+{a[0]}x-{a[0]*a[0]}}$

\n		\n

          (First, Outer, Inner, Last)

\n
$=$$\\simplify{x^2-{a[0]*a[0]}}$          (collect like terms)
\n

Method 3 (difference of two squares)

\n

Notice that the product will expand to be a difference of two squares. Square the first term minus the square of the second term. 

\n\n\n\n\n\n\n\n\n\n
$\\simplify{(x+{a[0]})(x-{a[0]})}$$=$\n

$\\simplify{x^2-{a[0]*a[0]}}$

\n		\n

          (difference of two squares)

\n
\n

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

$\\simplify{(x+{a[2]})^2}$ = [[0]]

\n

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "

Ensure you don't use brackets in your answer.

", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "x^2+{2*a[2]}x+{a[2]*a[2]}", "marks": 1, "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme"}], "steps": [{"prompt": "

It is important to realise that $\\simplify{(x+{a[2]})^2}=\\simplify{(x+{a[2]})(x+{a[2]})}$. Recall that squaring something is multiplying it by itself.

\n

 

\n

Method 1 (the distributive law)

\n

We expand $\\simplify{(x+{a[2]})(x+{a[2]})}$ 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{(x+{a[2]})(x+{a[2]})}$$=$\n

$\\simplify{x(x+{a[2]})+{a[2]}(x+{a[2]})}$

\n		\n

          (each term in one bracket times the entire other bracket)

\n
$=$$\\simplify{x^2+{a[2]}x+{a[2]}x+{a[2]*a[2]}}$          (use the distributive law on each bracket)
$=$$\\simplify{x^2+{2*a[2]}x+{a[2]*a[2]}}$          (collect like terms)
\n

Method 2 (FOIL)

\n

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{(x+{a[2]})(x+{a[2]})}$$=$\n

$\\simplify[basic]{x^2+{a[2]}x+{a[2]}x+{a[2]*a[2]}}$

\n		\n

          (First, Outer, Inner, Last)

\n
$=$$\\simplify{x^2+{2*a[2]}x+{a[2]*a[2]}}$          (collect like terms)
\n

Method 3 (perfect square)

\n

Notice that $\\simplify{(x+{a[2]})^2}$ is a perfect square. Square the first term, double the second term times the first, then square the last term, add them all together.

\n\n\n\n\n\n\n\n\n\n
$\\simplify{(x+{a[2]})}$$=$\n

$\\simplify{x^2+{2*a[2]}x+{a[2]*a[2]}}$

\n		\n

          (perfect square)

\n
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": "1", "prompt": "

$\\simplify{(w+{a[1]})(w-{a[1]})}$ = [[0]]

\n

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "

Ensure you don't use brackets in your answer.

", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": ["w"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "w^2-{a[1]*a[1]}", "marks": 1, "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme"}], "steps": [{"prompt": "

Method 1 (the distributive law)

\n

We expand $\\simplify{(w+{a[1]})(w-{a[1]})}$ 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{(w+{a[1]})(w-{a[1]})}$$=$\n

$\\simplify{w(w-{a[1]})+{a[1]}(w-{a[1]})}$

\n		\n

          (each term in one bracket times the entire other bracket)

\n
$=$$\\simplify{w^2-{a[1]}w+{a[1]}w-{a[1]*a[1]}}$          (use the distributive law on each bracket)
$=$$\\simplify{w^2-{a[1]*a[1]}}$          (collect like terms)
\n

Method 2 (FOIL)

\n

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{(w+{a[1]})(w-{a[1]})}$$=$\n

$\\simplify[basic]{w^2-{a[1]}w+{a[1]}w-{a[1]*a[1]}}$

\n		\n

          (First, Outer, Inner, Last)

\n
$=$$\\simplify{w^2-{a[1]*a[1]}}$          (collect like terms)
\n

Method 3 (difference of two squares)

\n

Notice that the product will expand to be a difference of two squares. Square the first term minus the square of the second term. 

\n\n\n\n\n\n\n\n\n\n
$\\simplify{(w+{a[1]})(w-{a[1]})}$$=$\n

$\\simplify{w^2-{a[1]*a[1]}}$

\n		\n

          (difference of two squares)

\n
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": "1", "prompt": "

$\\simplify{(r+{a[3]})(r+{a[3]})}$ = [[0]]

\n

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "

Ensure you don't use brackets in your answer.

", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": ["r"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "r^2+{2*a[3]}r+{a[3]*a[3]}", "marks": 1, "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme"}], "steps": [{"prompt": "

Method 1 (the distributive law)

\n

We expand $\\simplify{(r+{a[3]})(r+{a[3]})}$ 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{(r+{a[3]})(r+{a[3]})}$$=$\n

$\\simplify{r(r+{a[3]})+{a[3]}(r+{a[3]})}$

\n		\n

          (each term in one bracket times the entire other bracket)

\n
$=$$\\simplify{r^2+{a[3]}r+{a[3]}r+{a[3]*a[3]}}$          (use the distributive law on each bracket)
$=$$\\simplify{r^2+{2*a[3]}r+{a[3]*a[3]}}$          (collect like terms)
\n

Method 2 (FOIL)

\n

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{(r+{a[3]})(r+{a[3]})}$$=$\n

$\\simplify[basic]{r^2+{a[3]}r+{a[3]}r+{a[3]*a[3]}}$

\n		\n

          (First, Outer, Inner, Last)

\n
$=$$\\simplify{r^2+{2*a[3]}r+{a[3]*a[3]}}$          (collect like terms)
\n

Method 3 (perfect square)

\n

Notice that $\\simplify{(r+{a[3]})(r+{a[3]})}$ is a perfect square. Square the first term, double the second term times the first, then square the last term, add them all together.

\n\n\n\n\n\n\n\n\n\n
$\\simplify{(r+{a[3]})(r+{a[3]})}$$=$\n

$\\simplify{r^2+{2*a[3]}r+{a[3]*a[3]}}$

\n		\n

          (perfect square)

\n
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "

Expand and simplify the following.

", "variable_groups": [], "variablesTest": {"maxRuns": "127", "condition": ""}, "variables": {"a": {"definition": "shuffle(-12..12 except 0)[0..4]", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}}, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}