【数学I】例題1.1.1：多項式の整理と次数，定数項（One More）★

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% 例題I1.1.1：多項式の整理と次数，定数項（One More）★
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(1) 次の多項式を$x$について降べきの順に整理し，次数と定数項を求めよ．

\[
4-3x^3+2x-5x+x^3+x^2-7+3x^2
\]

(2) 次の多項式において，$[]$内の文字に着目したとき，その次数と定数項を求めよ．

\[
3a^2-4a^2b-ab^2+2b+5a^2b-5-2a,[a],[a\text{と}b]
\]

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(1)

\begin{align*}
&4-3x^3+2x-5x+x^3+x^2-7+3x^2\\
=&(-3x^3+x^3)+(x^2+3x^2)+(2x-5x)+(4-7)\\
=&-2x^3+4x^2-3x-3
\end{align*}

よって，次数は3，定数項は$-3$

(2) $a$に着目すると，

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

よって，$a$に着目したとき，次数は$2$，定数項は$2b-5$

また，$a$と$b$に着目すると，

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

よって，$a$と$b$に着目したとき，次数は$3$，定数項は$-5$

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% 問題I1.1.1：多項式の整理と次数，定数項（One More）★
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(1) 次の多項式を$x$について降べきの順に整理し，次数と定数項を求めよ．

\[
5+x^4-3x^3+2x-4x+3x^2-9-x^4
\]

(2) 次の多項式において，$[]$内の文字に着目したとき，その次数と定数項を求めよ．

\[
4b^2-3ab^2+ab-6a+7a^2-3+2b^3,[b],[a\text{と}b]
\]

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% 問題I1.1.1の解答（One More）★
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(1)

\begin{align*}
&5+x^4-3x^3+2x-4x+3x^2-9-x^4\\
=&(x^4-x^4)+(-3x^3)+(3x^2)+(2x-4x)+(5-9)\\
=&-3x^3+3x^2-2x-4
\end{align*}

よって，次数は3，定数項は$-4$

(2) $b$に着目すると，

\begin{align*}
4b^2-3ab^2+ab-6a+7a^2-3+2b^3&=(4-3a)b^2+ab+2b^3+7a^2-6a-3\\
&=2b^3+(4-3a)b^2+ab+7a^2-6a-3
\end{align*}

よって，$b$に着目したとき，次数は$3$，定数項は$7a^2-6a-3$

また，$a$と$b$に着目すると，

\[
4b^2-3ab^2+ab-6a+7a^2-3+2b^3=2b^3-3ab^2+ab+7a^2+4b^2-6a-3
\]

よって，$a$と$b$に着目したとき，次数は$3$，定数項は$-3$

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% 例題I1.1.1の側注：多項式の整理と次数，定数項（One More）★
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% 側注1
同類項をまとめ，降べきの順（次数の高い順）に整理する．

% 側注2
$-2x^{3}$より，最も高い次数は$3$である．

% 側注3
着目した文字以外の文字を定数として考える．

% 側注4
$a^2b$と$-ab^2$より，最も高い次数は$3$である（次数は掛け合わせた文字の総数）．

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% 問題I1.1.1の側注：多項式の整理と次数，定数項（One More）★
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% 側注1
同類項をまとめ，降べきの順（次数の高い順）に整理する．

% 側注2
$-3x^3$より，最も高い次数は3である．

% 側注3
着目した文字以外の文字を定数として考える．

% 側注4
$b^3$があるため，最も高い次数は$3$である（次数は掛け合わせた文字の総数）．

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% 例題I1.1.1の考え方・補足など：多項式の整理と次数，定数項（One More）★
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% 考え方
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多項式の次数$\cdots$同類項をまとめて整理した多項式において，最も次数が高い項の次数のこと．

(1) 同類項をまとめ，$x$について次数の高い順に並べる．

(2) 着目した文字以外の文字は，数として考える．例えば，$a$に着目したときは，$b$を数として扱う．

\[
\underbrace{3a^2}_{2\text{次}}-\underbrace{4a^2b}_{2\text{次}}-\underbrace{ab^2}_{1\text{次}}+\underbrace{2b}_{\text{定数項}}+\underbrace{5a^2b}_{2\text{次}}-\underbrace{5}_{\text{定数項}}-\underbrace{2a}_{1\text{次}}(a\text{に着目したとき})
\]

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文字に着目$\rightarrow$着目した文字以外の文字を数として考える．