【数学I】例題1.1.11：因数分解の工夫（次数の低い文字に着目）（One More）★★

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% 例題I1.1.11：因数分解の工夫（次数の低い文字に着目）（One More）★★
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次の式を因数分解せよ．

(1) $4a^2+2ab+b-1$

(2) $x^3+x^2y+2xy+y^2-1$

(3) $a^2+2ab+ac+b^2+bc$

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% 例題I1.1.11の解答（One More）★★
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(1)

\begin{align*}
4a^2+2ab+b-1&=(2a+1)b+4a^2-1\\
&=(2a+1)b+(2a+1)(2a-1)\\
&=(2a+1)(2a+b-1)
\end{align*}

(2)

\begin{align*}
&x^3+x^2y+2xy+y^2-1\\
=&y^2+(x^2+2x)y+(x^3-1)\\
=&y^2+(x^2+2x)y+(x-1)(x^2+x+1)\\
=&\{y+(x^2+x+1)\}\{y+(x-1)\}\\
=&(x^2+x+y+1)(x+y-1)
\end{align*}

(3)

\begin{align*}
&a^2+2ab+ac+b^2+bc\\
=&(a+b)c+(a^2+2ab+b^2)\\
=&(a+b)c+(a+b)^2\\
=&(a+b)\{c+(a+b)\}\\
=&(a+b)(a+b+c)
\end{align*}

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% 問題I1.1.11：因数分解の工夫（次数の低い文字に着目）（One More）★★
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次の式を因数分解せよ．

(1) $9x^2+3xy+y-1$

(2) $x^3+x^2y+3xy+y^2+2y-8$

(3) $xy+xz-y^2-z^2-2yz$

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% 問題I1.1.11の解答（One More）★★
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(1)

\begin{align*}
9x^2+3xy+y-1&=(3x+1)y+9x^2-1\\
&=(3x+1)y+(3x+1)(3x-1)\\
&=(3x+1)(3x+y-1)
\end{align*}

(2)

\begin{align*}
&x^3+x^2y+3xy+y^2+2y-8\\
=&y^2+(x^2+3x+2)y+(x^3-8)\\
=&y^2+(x^2+3x+2)y+(x-2)(x^2+2x+4)\\
=&\{y+(x^2+2x+4)\}\{y+(x-2)\}\\
=&(x^2+2x+y+4)(x+y-2)
\end{align*}

(3)

\begin{align*}
&xy+xz-y^2-z^2-2yz\\
=&(y+z)x-(y^2+2yz+z^2)\\
=&(y+z)x-(y+z)^2\\
=&(y+z)\{x-(y+z)\}\\
=&(y+z)(x-y-z)
\end{align*}

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% 例題I1.1.11の側注：因数分解の工夫（次数の低い文字に着目）（One More）★★
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% 側注1
$y$について整理する．

% 側注2
\begin{align*}
&a^3-b^3\\
=&(a-b)(a^2+ab+b^2)
\end{align*}

を用いて因数分解する．さらに，全体を$y$の2次式と考えて，たすき掛けを用いて因数分解する．

% 側注3
$b$について整理する．

% 側注4
$2a+1$をくくり出す．

% 側注5
$a^2+2ab+b^2=(a+b)^2$

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% 問題I1.1.11の側注：因数分解の工夫（次数の低い文字に着目）（One More）★★
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% 側注1
$y$について整理する．

% 側注2
\begin{align*}
&a^3-b^3\\
=&(a-b)(a^2+ab+b^2)
\end{align*}

を用いて因数分解する．さらに，全体を$y$の2次式と考えて，たすき掛けを用いて因数分解する．

% 側注3
$y$について整理する．

% 側注4
$3x+1$をくくり出す．

% 側注5
$a^2+2ab+b^2=(a+b)^2$

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% 例題I1.1.11の考え方・補足など：因数分解の工夫（次数の低い文字に着目）（One More）★★
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% 考え方
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一般に，式は次数が低いほど扱いやすい．例えば，(2)は次数の高い$x$に着目すると，

\[
x^3+x^2y+2xy+(y+1)(y-1)
\]

となり，因数分解の見通しが立てにくい．それに対して，次数の低い$y$に着目すると，

\[
y^2+(x^2+2x)y+x^3-1=y^2+(x^2+2x)y+(x-1)(x^2+x+1)
\]

となり，因数分解の見通しが立つ（全体を$y$の2次式と考えると，たすき掛けを用いた因数分解が適用できる形である）．

このように，複数の文字を含む式の因数分解は，次数の低い文字に着目するとよい．

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複数の文字を含む式の因数分解は，次数の低い文字に着目するとよい．