【数学I】例題1.1.10：因数分解（3次式）（One More）★★

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I1.1.10：因数分解（3次式）（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

次の式を因数分解せよ．

(1) $x^3+64$

(2) $8a^3-125b^3$

(3) $x^3+3x^2+3x+1$

(4) $x^3+2x^2-9x-18$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I1.1.10の解答（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(1)

\begin{align*}
x^3+64&=x^3+4^3\\
&=(x+4)\{x^2-x\cdot 4+4^2\}\\
&=(x+4)(x^2-4x+16)
\end{align*}

(2)

\begin{align*}
8a^3-125b^3&=(2a)^3-(5b)^3\\
&=(2a-5b)\{(2a)^2+2a\cdot 5b+(5b)^2\}\\
&=(2a-5b)(4a^2+10ab+25b^2)
\end{align*}

(3) $x^3+3x^2+3x+1=(x+1)^3$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I1.1.10の別解
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I1.1.10の解答（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(4)

\begin{align*}
x^3+2x^2-9x-18&=(x^3+2x^2)-(9x+18)\\
&=x^2(x+2)-9(x+2)\\
&=(x+2)(x^2-9)\\
&=(x+2)(x-3)(x+3)
\end{align*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I1.1.10：因数分解（3次式）（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

次の式を因数分解せよ．

(1) $x^3-27$

(2) $27m^3+8n^3$

(3) $x^3-6x^2+12x-8$

(4) $x^3-4x^2-x+4$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I1.1.10の解答（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(1)

\begin{align*}
x^3-27&=x^3-3^3\\
&=(x-3)\{x^2+x\cdot 3+3^2\}\\
&=(x-3)(x^2+3x+9)
\end{align*}

(2)

\begin{align*}
27m^3+8n^3&=(3m)^3+(2n)^3\\
&=(3m+2n)\{(3m)^2-3m\cdot 2n+(2n)^2\}\\
&=(3m+2n)(9m^2-6mn+4n^2)
\end{align*}

(3) $x^3-6x^2+12x-8=(x-2)^3$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I1.1.10の別解
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{align*}
x^3-6x^2+12x-8&=(x^3-8)+(-6x^2+12x)\\
&=(x-2)(x^2+2x+4)-6x(x-2)\\
&=(x-2)\{x^2+2x+4-6x\}\\
&=(x-2)(x^2-4x+4)\\
&=(x-2)(x-2)^2\\
&=(x-2)^3
\end{align*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I1.1.10の解答（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(4)

\begin{align*}
x^3-4x^2-x+4&=(x^3-4x^2)-(x-4)\\
&=x^2(x-4)-1(x-4)\\
&=(x-4)(x^2-1)\\
&=(x-4)(x-1)(x+1)
\end{align*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I1.1.10の側注：因数分解（3次式）（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 側注1
\begin{align*}
&a^3+b^3\\
=&(a+b)(a^2-ab+b^2)
\end{align*}

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

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

% 側注4
組み合わせを工夫する．

% 側注5
$x^3+1=x^3+1^3$

% 側注6
$x+1$をくくり出す．

% 側注7
$x+2$をくくり出す．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I1.1.10の側注：因数分解（3次式）（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 側注1
\begin{align*}
&a^3+b^3\\
=&(a+b)(a^2-ab+b^2)
\end{align*}

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

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

% 側注4
組み合わせを工夫する．

% 側注5
$x^3-8=x^3-2^3$

% 側注6
$x-2$をくくり出す．

% 側注7
$x-4$をくくり出す．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I1.1.10の考え方・補足など：因数分解（3次式）（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 考え方
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

3次式の因数分解を用いるときは，符号に注意すること．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 3次式の因数分解の公式
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\[
a^3+b^3=(a+b)(a^2-ab+b^2),a^3-b^3=(a-b)(a^2+ab+b^2)
\]

\[
a^3+3a^2b+3ab^2+b^3=(a+b)^3,a^3-3a^2b+3ab^2-b^3=(a-b)^3
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 補足
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

なお，(4)のように公式が使えない場合は，組み合わせを工夫して共通因数のくくり出しを考えるとよい．