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因数分解のまとめ | 夢を叶える塾
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因数分解のまとめ

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数学に関する投稿の第2回目です。
こちらは単元のポイントなどをまとめた記事になります。

因数分解の特徴

整式の演算同様に数学の超基礎です。
整式の演算のようにルールに沿ってルーチンワークを行うものではないため、気付きが必要なケースが出てきます。
しかし、演習量を積むことで気付きが起きやすくなり、また、気付きまでの経路も増加します。

何がうれしいのか

「a2+2ab+b2=(a+b)2
という因数分解ができると、
「(a2+2ab+b2)/(a+b)=(a+b)2/(a+b)=(a+b)」
という式変形ができます。また、
「x2-a2=(x+a)(x-a)」
という因数分解ができると、
「x2-a2=0」
⇒「左辺=(x-a)(x+a)=0」
⇒「x=a,-aが解」
となり、方程式の解を求めることにもつながります。

解き方の基本(1)公式を使う

「a2+2ab+b2=(a+b)2
「a3+b3=(a+b)(a2-ab+b2)」
等の公式を覚えることで対応できる問題も多くあります。

解き方の基本(2)因数定理

「x2-a2
⇒「x=a,-aを代入すると与式=0」
⇒「与式=(x-a)(x+a)」
三次式の因数分解では、一つの因数を見つけると、残りの二次式の因数分解に帰着できます。
三次式以上の因数分解では定数項を素因数分解した素数の組み合わせを代入すると因数が見つかることもしばしばあります。

解き方の+α

上記の解き方以外に、解法の方針として、3点あげておきたいと思います。

共通因数

「a2」など二次式がいない、「ab+a+b+1」のようなケースが該当します。
「ab+a+b+1=a(b+1)+(b+1)=(a+1)(b+1)」

降べきの順に整理

単項式の最大指数が単項式の数より多いような「a2+2ab+b2+2a+2b+1」のようなケースが該当します。
「a2+2ab+b2+2a+2b+1
 =a2+2ab+2a+b2+2b+1
 =a2+2a(b+1)+(b+1)2
 =(a+(b+1))(a+(b+1))
 =(a+b+1)2

単項式の指数をチェックする

「ab+a+b+1」は、「2」、「1」、「1」、「0」なので、「1」と「0」の和同士の積であると考えられます。
「a2+2ab+b2+2a+2b+1」も「2」、「2」、「2」、「1」、「1」、「0」なので、「1」と「0」の和同士の積であると考えられます。また、「2」が3個ありますので、少なくとも一方が「1」、「1」、「0」の形であることが予想できます。

まとめ

整式の演算同様、数学のベースになります。
演習問題では正答率9割以上を目指したいところです。
単純な計算ミスを防止できるよう、展開し直して元の式に戻る検算も身に着けておくと良いでしょう。

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