【数学I】例題1.1.3：多項式の乗法（One More）★

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% 例題I1.1.3：多項式の乗法（One More）★
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次の計算をせよ．

(1) $4x^3y^2\times(-2xy^3)^2$

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

(3) $(x+2)(2x^2-x+5)$

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

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% 例題I1.1.3の解答（One More）★
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(1)

\begin{align*}
4x^3y^2\times(-2xy^3)^2&=4x^3y^2\times(-2)^2x^2(y^3)^2=4x^3y^2\times 4x^2y^6\\
&=4\cdot 4x^{3+2}y^{2+6}=16x^5y^8
\end{align*}

(2)

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

(3)

\begin{align*}
(x+2)(2x^2-x+5)&=x(2x^2-x+5)+2(2x^2-x+5)\\
&=2x^3-x^2+5x+4x^2-2x+10\\
&=2x^3+3x^2+3x+10
\end{align*}

(4)

\begin{align*}
&(2x^3-4x^2+x)(2-x+x^2)\\
=&(2x^3-4x^2+x)(x^2-x+2)\\
=&2x^3(x^2-x+2)-4x^2(x^2-x+2)+x(x^2-x+2)\\
=&2x^5-2x^4+4x^3-4x^4+4x^3-8x^2+x^3-x^2+2x\\
=&2x^5-6x^4+9x^3-9x^2+2x
\end{align*}

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% 問題I1.1.3：多項式の乗法（One More）★
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次の計算をせよ．

(1) $3x^2y\times(-4xy^2)^2$

(2) $5abc^2(2a^2-3b+4c)$

(3) $(x-3)(x^2+4x-7)$

(4) $(x^3-2x+5)(3x^2-x+4)$

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% 問題I1.1.3の解答（One More）★
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(1)

\begin{align*}
3x^2y\times(-4xy^2)^2&=3x^2y\times(-4)^2x^2(y^2)^2=3x^2y\times 16x^2y^4\\
&=3\cdot 16x^{2+2}y^{1+4}=48x^4y^5
\end{align*}

(2)

\begin{align*}
5abc^2(2a^2-3b+4c)&=5abc^2\cdot 2a^2+5abc^2\cdot(-3b)+5abc^2\cdot 4c\\
&=10a^3bc^2-15ab^2c^2+20abc^3
\end{align*}

(3)

\begin{align*}
(x-3)(x^2+4x-7)&=x(x^2+4x-7)-3(x^2+4x-7)\\
&=x^3+4x^2-7x-3x^2-12x+21\\
&=x^3+x^2-19x+21
\end{align*}

(4)

\begin{align*}
&(x^3-2x+5)(3x^2-x+4)\\
=&x^3(3x^2-x+4)-2x(3x^2-x+4)+5(3x^2-x+4)\\
=&3x^5-x^4+4x^3-6x^3+2x^2-8x+15x^2-5x+20\\
=&3x^5-x^4-2x^3+17x^2-13x+20
\end{align*}

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% 例題I1.1.3の側注：多項式の乗法（One More）★
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% 側注1
$(-2xy^3)^2$を先に計算する．

% 側注2
係数の積，文字の積をそれぞれ計算する．

% 側注3
$A(B+C)=AB+AC$より，次のことがいえる．

\[
A(B+C+D)=AB+AC+AD
\]

% 側注4
\begin{align*}
&(A+B)(C+D+E)\\
&=AC+AD+AE\\
&+BC+BD+BE
\end{align*}

% 側注5
降べきの順に整理してから展開すると，まとめやすくなる．

% 側注6
降べきの順に整理する．

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% 問題I1.1.3の側注：多項式の乗法（One More）★
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% 側注1
$(-4xy^2)^2$を先に計算する．

% 側注2
係数の積，文字の積をそれぞれ計算する．

% 側注3
$A(B+C)=AB+AC$より，次のことがいえる．

\[
A(B+C+D)=AB+AC+AD
\]

% 側注4
\begin{align*}
&(A+B)(C+D+E)\\
&=AC+AD+AE\\
&+BC+BD+BE
\end{align*}

% 側注5
降べきの順に整理する．

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% 例題I1.1.3の考え方・補足など：多項式の乗法（One More）★
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% 考え方
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(1) 指数法則を用いる．

\[
a^m\times a^n=a^{m+n},(a^m)^n=a^{mn},(ab)^n=a^nb^n
\]

(2) 分配法則を用いる．

\[
A(B+C)=AB+AC
\]

(3) 分配法則を用いて，次のように展開する．

\[
(x+3)(x^2+2x-5)
\]

(4) 降べきの順に整理してから展開する．

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% One Point
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指数法則，分配法則を用いて計算する．

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% 注意
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$(-a)^n$は符号に注意して計算する．

$n$が偶数のとき，$(-a)^n=a^n$であり，$n$が奇数のとき，$(-a)^n=-a^n$である．