// Numbas version: exam_results_page_options {"name": "Musa's copy of Expand brackets", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Musa's copy of Expand brackets", "tags": [], "metadata": {"description": "

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

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

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

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

", "advice": "

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

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.

a)

\\[
\\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}
\\]

b)

\\[
\\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}
\\]

c)

\\[
\\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}
\\]

d)

\\[
\\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}
\\]

e)

\\[
\\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}
\\]

f)

\\[
\\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}
\\]

g)

\\[
\\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}
\\]

h)

\\[
\\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}
\\]

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random variables for part 1

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$\\var{c[0]}x+\\var{c[1]}x+\\var{c[2]}x=$ [[0]]

You must collect like terms to fully simplify.

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$\\var{a[1]}x^2+\\var{a[2]}x^2+\\var{a[3]}x+\\var{a[4]}x +\\var{a[0]}=$ [[0]]

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$\\var{b[0]}y^5+\\var{b[1]}y^5+\\var{b[2]}y^5+\\var{b[4]}y^5+\\var{b[3]}y^5=$ [[0]]