// Numbas version: exam_results_page_options {"name": "Expand brackets and collect like terms - two questions", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"preventleave": false, "allowregen": true, "showfrontpage": false}, "question_groups": [{"questions": [{"type": "question", "tags": [], "contributors": [{"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/", "name": "Christian Lawson-Perfect"}, {"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/315/", "name": "Angharad Thomas"}, {"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/", "name": "Bradley Bush"}, {"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/", "name": "Aiden McCall"}], "name": "Expand brackets and collect like terms - two questions", "ungrouped_variables": [], "preamble": {"js": "", "css": ""}, "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "rulesets": {}, "variable_groups": [{"name": "B group", "variables": ["a1", "b1", "c1"]}, {"name": "Part a", "variables": ["a", "b", "c", "d", "f", "g", "h", "j"]}], "advice": "

When simplifying expressions, only terms of the same type or like terms can be added together.

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Algebraic symbols or letters can be added together provided that they are raised to the same power. For example, we can add $x^2+x^2=2x^2$, but we cannot collect both $x^2$ and $x$ into one term.

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#### a)

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\\\begin{align} \\var{c[0]}x+\\var{c[1]}x+\\var{c[2]}x&=(\\var{c[0]}+\\var{c[1]}+\\var{c[2]})x\\\\ &=\\simplify{({c[0]}+{c[1]}+{c[2]})}x \\end{align} \

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#### b)

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\\\begin{align} \\var{a[1]}x^2+\\var{a[2]}x^2+\\var{a[3]}x+\\var{a[4]}x +\\var{a[0]}&=(\\var{a[1]}+\\var{a[2]})x^2+(\\var{a[3]}+\\var{a[4]})x +\\var{a[0]}\\\\ &=\\simplify{({a[1]}+{a[2]})}x^2+\\simplify{({a[3]}+{a[4]})}x+\\var{a[0]} \\end{align} \

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#### c)

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\\\begin{align} \\var{b[0]}y^5+\\var{b[1]}y^5+\\var{b[2]}y^5+\\var{b[4]}y^5+\\var{b[3]}y^5&=(\\var{b[0]}+\\var{b[1]}+\\var{b[2]}+\\var{b[4]}+\\var{b[3]})y^5\\\\ &=\\simplify{({b[1]}+{b[2]}+{b[3]}+{b[4]}+{b[0]})}y^5 \\end{align} \

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#### d)

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\\\begin{align} \\var{d[0]}ab+\\var{d[1]}abc+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c+\\var{d[5]}abc &=(\\var{d[1]}+\\var{d[5]})abc+\\var{d[0]}ab+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c\\\\ &=\\simplify{{d[1]}+{d[5]}}abc+\\var{d[0]}ab+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c \\end{align} \

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#### e)

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\\\begin{align} \\var{f[0]}a^2b+\\var{f[1]}ab^2+\\var{f[2]}ab+\\var{f[3]}a^2b+\\var{f[4]}ab^2 &=(\\var{f[0]}+\\var{f[3]})a^2b+(\\var{f[1]}+\\var{f[4]})ab^2+\\var{f[2]}ab\\\\ &=\\simplify{{f[0]}+{f[3]}}a^2b+\\simplify{{f[1]}+{f[4]}}ab^2+\\var{f[2]}ab \\end{align} \

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#### f)

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\\\begin{align} \\var{g[0]}(\\var{g[1]}x+\\var{g[2]}y)+\\var{g[4]}x+\\var{g[5]}y &=(\\var{g[0]}\\times \\var{g[1]}+\\var{g[4]})x+(\\var{g[0]} \\times\\var{g[2]}+\\var{g[5]})y\\\\ &=(\\simplify{{g[0]}*{g[1]}}+\\var{g[4]})x+(\\simplify{{g[0]}*{g[2]}}+\\var{g[5]})y\\\\ &=\\simplify{{g[0]}*{g[1]}+{g[4]}}x+\\simplify{{g[0]}*{g[2]}+{g[5]}}y \\end{align} \

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#### g)

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\\\begin{align} \\var{h[0]}x(\\var{h[1]}x+\\var{h[2]}z)+\\var{h[3]}x+\\var{h[6]}z+\\var{h[4]}x^2+\\var{h[5]}z^2 &=(\\simplify[]{{h[0]}{h[1]}}+\\var{h[4]})x^2+(\\simplify[]{{h[0]}{h[2]}})zx+\\var{h[3]}x+\\var{h[5]}z^2+\\var{h[6]}z\\\\ &=(\\simplify{{h[0]}{h[1]}}+\\var{h[4]})x^2+(\\simplify[]{{h[0]}{h[2]}})zx+\\var{h[3]}x+\\var{h[5]}z^2+\\var{h[6]}z\\\\ &=\\simplify{{h[0]}*{h[1]}+{h[4]}}x^2+\\simplify{{h[0]}*{h[2]}}zx+\\simplify{{h[3]}x+{h[5]}}z^2+\\var{h[6]}z \\end{align} \

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#### h)

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\\\begin{align} \\var{j[0]}(\\var{j[1]}x-\\var{j[2]}y)+\\var{j[3]}(\\var{j[4]}x-\\var{j[5]}y)+\\var{j[6]}(\\var{j[7]}x-\\var{j[8]}y) &= (\\simplify[]{{j[0]}{j[1]}}+\\simplify[]{{j[3]}{j[4]}}+\\simplify[]{{j[6]}{j[7]}})x-(\\simplify[]{{j[0]}{j[2]}}+\\simplify[]{{j[3]}{j[5]}}+\\simplify[]{{j[6]}{j[8]}})y\\\\ &= (\\simplify{{j[0]}{j[1]}}+\\simplify{{j[3]}{j[4]}}+\\simplify{{j[6]}{j[7]}})x-(\\simplify{{j[0]}{j[2]}}+\\simplify{{j[3]}{j[5]}}+\\simplify{{j[6]}{j[8]}})y\\\\ &= \\simplify{({j[0]}*{j[1]}+{j[4]*j[3]}+{j[6]}*{j[7]})x}-\\simplify{({j[0]}*{j[2]}+{j[5]}{j[3]}+{j[6]}*{j[8]})y} \\end{align} \

\n", "statement": "

For each expression below, collect like terms and expand brackets.

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The * symbol is required between algebraic symbols, e.g. $5ab^2$ should be written 5*a*b^2.

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$\\var{j[0]}(\\var{j[1]}x-\\var{j[2]}y)+\\var{j[3]}(\\var{j[4]}x-\\var{j[5]}y)+\\var{j[6]}(\\var{j[7]}x-\\var{j[8]}y)=$ [[0]]

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random variables for part 1

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Eight expressions, of increasing complexity. The student must simplify them by expanding brackets and collecting like terms.

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