【数学I】例題1.1.8：因数分解の基本（One More）★

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I1.1.8：因数分解の基本（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

次の式を因数分解せよ．

(1) $3x^3y+9x^2y+15xy^3$

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

(3) $x^2+10x+25$

(4) $9a^3-6a^2+a$

(5) $16x^2-(x+1)^2$

(6) $a^2+5a+6$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I1.1.8の解答（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(1) $3x^3y+9x^2y+15xy^3=3xy(x^2+3x+5y^2)$

(2) $xy-y-x+1=(x-1)y-(x-1)=(x-1)(y-1)$

(3) $x^2+10x+25=x^2+2\cdot x\cdot 5+5^2=(x+5)^2$

(4) $9a^3-6a^2+a=a(9a^2-6a+1)=a\{(3a)^2-2\cdot 3a\cdot 1+1^2\}=a(3a-1)^2$

(5)

\begin{align*}
16x^2-(x+1)^2&=\{4x+(x+1)\}\{4x-(x+1)\}\\
&=(4x+x+1)(4x-x-1)\\
&=(5x+1)(3x-1)
\end{align*}

(6) $a^2+5a+6=a^2+(2+3)a+2\cdot 3=(a+2)(a+3)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I1.1.8：因数分解の基本（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

次の式を因数分解せよ．

(1) $4x^3y^2+8x^2y^2+12xy^3$

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

(3) $x^2-14x+49$

(4) $25m^3-20m^2+4m$

(5) $9y^2-(y-2)^2$

(6) $x^2+7x+10$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I1.1.8の解答（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(1) $4x^3y^2+8x^2y^2+12xy^3=4xy^2(x^2+2x+3y)$

(2) $ab-b-a+1=(a-1)b-(a-1)=(a-1)(b-1)$

(3) $x^2-14x+49=x^2-2\cdot x\cdot 7+7^2=(x-7)^2$

(4)

\begin{align*}
25m^3-20m^2+4m&=m(25m^2-20m+4)\\
&=m\{(5m)^2-2\cdot 5m\cdot 2+2^2\}\\
&=m(5m-2)^2
\end{align*}

(5)

\begin{align*}
9y^2-(y-2)^2&=\{3y+(y-2)\}\{3y-(y-2)\}\\
&=(3y+y-2)(3y-y+2)\\
&=(4y-2)(2y+2)\\
&=4(2y-1)(y+1)
\end{align*}

(6) $x^2+7x+10=x^2+(2+5)x+5\cdot 2=(x+2)(x+5)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I1.1.8の側注：因数分解の基本（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 側注1
$3,9,15$の最大公約数は$3$である．

% 側注2
$y$について整理すると，共通因数$x-1$が見つかる．

% 側注3
$a^2+2ab+b^2=(a+b)^2$

% 側注4
先に$a$をくくり出す．

% 側注5
括弧を外すときは符号に注意すること．

% 側注6
和が$5$，積が$6$になる数を探す．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I1.1.8の側注：因数分解の基本（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 側注1
$4,8,12$の最大公約数は$4$である．

% 側注2
$b$について整理すると，共通因数$a-1$が見つかる．

% 側注3
$a^2-2ab+b^2=(a-b)^2$

% 側注4
先に$m$をくくり出す．

% 側注5
括弧を外すときは符号に注意すること．

% 側注6
和が$7$，積が$10$になる数を探す．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I1.1.8の考え方・補足など：因数分解の基本（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

1つの多項式について，多項式の積の形に変形することを，因数分解するという．因数分解は，展開の逆の操作である．(3)以降は，因数分解の公式の利用を考える．

(1) 共通因数のくくり出しを考える．

(2) $-x+1=-(x-1)$と変形する．

(3) $a^2+2ab+b^2=(a+b)^2$

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

(5) $a^2-b^2=(a+b)(a-b)$

(6) $x^2+(a+b)x+ab=(x+a)(x+b)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 因数分解の公式
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(i) $a^2+2ab+b^2=(a+b)^2$

(ii) $a^2-2ab+b^2=(a-b)^2$

(iii) $a^2-b^2=(a+b)(a-b)$

(iv) $x^2+(a+b)x+ab=(x+a)(x+b)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 注意
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

因数分解は，先にくくり出しを試みる．次に，因数分解の公式の利用を考えるとよい．