// Numbas version: exam_results_page_options {"name": "Terry's copy of Expanding a binomial product (monic factors)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "", "statement": "

Expand and simplify the following.

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Ensure you don't use brackets in your answer.

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Method 1 (the distributive law)

\n

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

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

\n
\n

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

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

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

\n
\n

          (First, Outer, Inner, Last)

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

\n

\n

", "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": ""}, {"showCorrectAnswer": true, "gaps": [{"checkingType": "absdiff", "notallowed": {"message": "

Ensure you don't use brackets in your answer.

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Method 1 (the distributive law)

\n

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

$\\simplify{x(x+{b[1]})+{a[1]}(x+{b[1]})}$

\n
\n

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

\n
$=$$\\simplify{x^2+{b[1]}x+{a[1]}x+{a[1]*b[1]}}$          (use the distributive law on each bracket)
$=$$\\simplify{x^2+{b[1]+a[1]}x+{a[1]*b[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{(x+{a[1]})(x+{b[1]})}$$=$\n

$\\simplify[basic]{x^2+{b[1]}x+{a[1]}x+{a[1]*b[1]}}$

\n
\n

          (First, Outer, Inner, Last)

\n
$=$$\\simplify{x^2+{b[1]+a[1]}x+{a[1]*b[1]}}$          (collect like terms)
", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "type": "information", "showFeedbackIcon": true, "variableReplacements": [], "marks": 0, "customMarkingAlgorithm": ""}], "marks": 0, "stepsPenalty": "1", "unitTests": [], "prompt": "

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

\n

\n

", "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": ""}, {"showCorrectAnswer": true, "gaps": [{"checkingType": "absdiff", "notallowed": {"message": "

Ensure you don't use brackets in your answer.

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Method 1 (the distributive law)

\n

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

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

\n
\n

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

\n
$=$$\\simplify{m^2+{b[2]}m+{a[2]}m+{a[2]*b[2]}}$          (use the distributive law on each bracket)
$=$$\\simplify{m^2+{b[2]+a[2]}m+{a[2]*b[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{(m+{a[2]})(m+{b[2]})}$$=$\n

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

\n
\n

          (First, Outer, Inner, Last)

\n
$=$$\\simplify{m^2+{b[2]+a[2]}m+{a[2]*b[2]}}$          (collect like terms)
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$\\simplify{(m+{a[2]})(m+{b[2]})}$ = [[0]]

\n

\n

", "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": ""}, {"showCorrectAnswer": true, "gaps": [{"checkingType": "absdiff", "notallowed": {"message": "

Ensure you don't use brackets in your answer.

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Method 1 (the distributive law)

\n

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

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

\n
\n

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

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

\n

          (collect like terms)

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

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

\n
\n

          (First, Outer, Inner, Last)

\n
$=$$\\simplify{t^2+{b[3]+a[3]}t+{a[3]*b[3]}}$          (collect like terms)
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$\\simplify{(t+{a[3]})(t+{b[3]})}$ = [[0]]

\n

\n

", "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": ""}], "extensions": [], "tags": ["binomial", "binomial product", "distributive law", "expanding", "Factorisation", "factorisation", "Factors", "factors", "monic", "quadratic"], "rulesets": {}, "ungrouped_variables": ["a", "b"], "type": "question", "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/"}]}