展開する (K(s+a))/(s+1)*1/(s(s+2)(s+3))

解

展開する

\frac{K ( s + a )}{s + 1} \cdot \frac{1}{s ( s + 2 ) ( s + 3 )}

\frac{Ks + a K}{s ^{4} + 6 s ^{3} + 11 s ^{2} + 6 s}

解答ステップ

\frac{K ( s + a )}{s + 1} \cdot \frac{1}{s ( s + 2 ) ( s + 3 )}

\frac{K ( s + a )}{s + 1} = \frac{Ks + a K}{s + 1}

= \frac{Ks + a K}{s + 1} \cdot \frac{1}{s ( s + 2 ) ( s + 3 )}

\frac{1}{s ( s + 2 ) ( s + 3 )} = \frac{1}{s ^{3} + 5 s ^{2} + 6 s}

= \frac{Ks + a K}{s + 1} \cdot \frac{1}{s ^{3} + 5 s ^{2} + 6 s}

分数を乗じる:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{( Ks + a K ) \cdot 1}{( s + 1 ) ( s ^{3} + 5 s ^{2} + 6 s )}

(Ks + a K) \cdot 1 = Ks + a K

= \frac{Ks + a K}{( s + 1 ) ( s ^{3} + 5 s ^{2} + 6 s )}

拡張

(s + 1) (s^{3} + 5 s^{2} + 6 s) : s^{4} + 6 s^{3} + 11 s^{2} + 6 s

= \frac{Ks + a K}{s ^{4} + 6 s ^{3} + 11 s ^{2} + 6 s}