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Simplify the following by collecting like terms.

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$\\simplify[!collectnumbers]{{a[1]}x+{b[1]}y+{c[1]}z+{b[2]}y+{a[2]}x+{c[2]}z+{a[0]}x+{c[0]}z+{b[0]}y}$ = [[0]]

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Like terms are terms where the variable part is the same. For example, $4x$ and $-x$ have the same variable part $x$. However, $3x$ and $-2y$ have different variable parts and are therefore unlike terms (or not like terms).

We can only collect like terms! Just like we can't say 2 m + 3 cm equals 5 m or 5 cm, we can't say $2x+3y$ equals $5x$ or $5y$! We can, however, say $2a+3a=5a$.

In our question we look at all the terms with a variable part of $x$ and add up all the corresponding coefficients, we do the same for the $y$ terms and the $z$ terms:

\\[\\begin{align}
&\\simplify[!collectnumbers]{{a[1]}x+{b[1]}y+{c[1]}z+{b[2]}y+{a[2]}x+{c[2]}z+{a[0]}x+{c[0]}z+{b[0]}y}\\\\
&=\\simplify[basic]{({a[1]}+{a[2]}+{a[0]})x+({b[1]}+{b[2]}+{b[0]})y+({c[1]}+{c[2]}+{c[0]})z}\\end{align}\\]

We present this as the sum of three unlike terms:

\\[\\simplify{{sum(a)}x+{sum(b)}y+{sum(c)}z}\\]

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$\\simplify[!collectnumbers]{{d[1]}x^2+{f[1]}x+{g[1]}+{d[0]}x^2+{f[0]}x+{g[0]}}$ = [[0]]

Like terms are terms where the variable part is the same. For example, $4x$ and $-x$ have the same variable part $x$. However, $3x^2$ and $-2x$ have different variable parts and are therefore unlike terms (or not like terms).

We can only collect like terms! Just like we can't say 2 m + 3 cm equals 5 m or 5 cm, we can't say $2x^2+3x$ equals $5x^2$ or $5x$! We can, however, say $2x^2+3x^2=5x^2$.

In our question we look at all the terms with a variable part of $x^2$ and add up all the corresponding coefficients (the numbers in front of the variables), we do the same for the $x$ terms and the constant terms (the terms with no variable part):

\\[\\begin{align}&\\simplify[!collectnumbers]{{d[1]}x^2+{f[1]}x+{g[1]}+{d[0]}x^2+{f[0]}x+{g[0]}}\\\\&=\\simplify[basic]{({d[1]}+{d[0]})x^2+({f[1]}+{f[0]})x+({g[1]}+{g[0]})}\\end{align}\\]

We present this as the sum of three unlike terms:

\\[\\simplify[!noleadingminus, basic]{{sum(d)}x^2+{sum(f)}x+{sum(g)}}\\]

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