Giải bài 8 trang 143 – SGK môn Giải tích lớp 12
Thực hiện các phép tính sau:
a) \(\left( 3+2i \right)\left[ \left( 2-i \right)+\left( 3-2i \right) \right]\);
b) \(\left( 4-3i \right)+\dfrac{1+i}{2+i}\);
c) \({{\left( 1+i \right)}^{2}}-{{\left( 1-i \right)}^{2}}\);
d) \(\dfrac{3+i}{2+i}-\dfrac{4-3i}{2-i}\).
a) \(\left( 3+2i \right)\left[ \left( 2-i \right)+\left( 3-2i \right) \right]\)
\(\begin{aligned} & =\left( 3+2i \right)\left( 5-3i \right) \\ & =15+6+\left( -9+10 \right)i \\ & =21+i \\ \end{aligned} \)
b) \(\left( 4-3i \right)+\dfrac{1+i}{2+i}\)
\(\begin{aligned} & =4-3i+\dfrac{\left( 1+i \right)\left( 2-i \right)}{5} \\ & =4-3i+\dfrac{3}{5}+\dfrac{1}{5}i \\ & =\dfrac{23}{5}-\dfrac{14}{5}i \\ \end{aligned} \)
c) \({{\left( 1+i \right)}^{2}}-{{\left( 1-i \right)}^{2}}\)
\(\begin{aligned} & =1+2i-1-\left( 1-2i-1 \right) \\ & =4i \\ \end{aligned} \)
d) \(\dfrac{3+i}{2+i}-\dfrac{4-3i}{2-i}\)
\(\begin{aligned} & =\dfrac{\left( 3+i \right)\left( 2-i \right)}{5}-\dfrac{\left( 4-3i \right)\left( 2+i \right)}{5} \\ & =\dfrac{7}{5}-\dfrac{1}{5}i-\dfrac{11}{5}+\dfrac{2}{5}i \\ & =-\dfrac{4}{5}+\dfrac{1}{5}i \\ \end{aligned} \)
Ghi nhớ:
Cho hai số phức \(z=a+bi\) và \(z'=a'+b'i\)
Ta có:
\(z\pm z'=(a\pm a')+(b'\pm b')i\)
\(z.z'=(aa'-bb')+(ab'+ba')i\)
\(\dfrac{z}{z'}=\dfrac{z.\overline z'}{|z'|^2}|\)