// Numbas version: exam_results_page_options {"name": "1.3.3. Removing brackets from expressions of the form (a+b)(c+d)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "1.3.3. Removing brackets from expressions of the form (a+b)(c+d)", "tags": [], "metadata": {"description": "

Removing brackets from expressions of the form (a+b)(c+d). Part of HELM Book 1.3

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Removing brackets from expressions of the form $(a+b)(c+d)$

Sometimes it is necessary to consider two bracketed terms multiplied together. In the expression $(a+b)(c+d)$, by regarding the first bracket as a single term we can use the result in Key Point 14 to write it as $(a+b)c + (a+b)d$. Removing the brackets from each of these terms produces $ac + bc + ad + bd$. More concisely:

Key Point 16

\$(a+b)(c+d)=(a+b)c+(a+b)d = ac + bc + ad + bd\$

We see that each term in the first bracketed expression multiplies each term in the second bracketed expression.

Example 38

Remove the brackets from $(3+x)(2+y)$

Solution

We find $(3 + x)(2 + y) = (3 + x)(2) + (3 + x)y$

$= (3)(2) + (x)(2) + (3)(y) + (x)(y) = 6 + 2x + 3y + xy$

Example 39

Remove the brackets from $(3x + 4)(x + 2)$ and simplify your result.

Solution

$(3x + 4)(x + 2) = (3x + 4)(x) + (3x + 4)(2)$

$= 3x^2 + 4x + 6x + 8 = 3x^2 + 10x + 8$

Example 40

Remove the brackets from $(a+b)^2$ and simplify your result.

Solution

When a quantity is squared it is multiplied by itself. Thus

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

= a^2 + ba + ab + b^2 = a^2 + 2ab + b^2$

Key Point 17

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

\$(a − b)^2 = a^2 − 2ab + b^2 \$

", "advice": "", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"q1expr": {"name": "q1expr", "group": "Ungrouped variables", "definition": "simplify(expression(\"(x+\" + q1consts[0] + \")*(x+\" + q1consts[1] + \")\"),[\"basic\"])", "description": "", "templateType": "anything", "can_override": false}, "q1ans": {"name": "q1ans", "group": "Ungrouped variables", "definition": "simplify(\nexpression(\"x^2 +(\" + q1consts[0]+ \"+\" + q1consts[1] + \")*x+\" + q1consts[0]+ \"*\" + q1consts[1]),\n [\"all\"])", "description": "", "templateType": "anything", "can_override": false}, "q1consts": {"name": "q1consts", "group": "Ungrouped variables", "definition": "repeat(random(-9..9 except 0),2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["q1expr", "q1ans", "q1consts"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": true, "customName": "Task 1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Remove the brackets from $\\var{q1expr}$ and simplify the result.

", "answer": " {q1ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": ""}, "valuegenerators": []}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Example 41

Explain the distinction between $(x + 3)(x + 2)$ and $x + 3(x + 2)$.

Solution

In the first expression removing the brackets we find

$(x + 3)(x + 2) = x^2 + 3x + 2x + 6$

$= x^2 + 5x + 6$

In the second expression we have

$x + 3(x + 2) = x + 3x + 6 = 4x + 6$

Note that in the second expression the term (x + 2) is only multiplied by 3 and not by x.

Example 42

Remove the brackets from $(s^2 + 2s + 4)(s + 3)$.

Solution

Each term in the first bracket must multiply each term in the second. Working through all combinations systematically we have

$(s^2 + 2s + 4)(s + 3) = (s^2 + 2s + 4)(s) + (s^2 + 2s + 4)(3)$

$= s^3 + 2s^2 + 4s + 3s^2 + 6s + 12$

$= s^3 + 5s^2 + 10s + 12$

"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}]}]}], "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}]}