How do you simplify (3-i)(1+2i)(2+3i)?

Mar 20, 2016

$\left(3 - i\right) \left(1 + 2 i\right) \left(2 + 3 i\right) = - 5 + 25 i$

Explanation:

Multiply out one pair of numbers as a product of binomials, simplify using ${i}^{2} = - 1$ and repeat with the remaining pair.

You can use FOIL if you find it helpful.

$\left(3 - i\right) \left(1 + 2 i\right)$

$= {\overbrace{\left(3 \cdot 1\right)}}^{\text{First" + overbrace((3*2i))^"Outside" + overbrace((-i*1))^"Inside" + overbrace((-i * 2i))^"Last}}$

$= 3 + 6 i - i - 2 {i}^{2}$

$= 3 + 5 i + 2$

$= 5 + 5 i$

Similarly:

$\left(5 + 5 i\right) \left(2 + 3 i\right) = 10 + 15 i + 10 i - 15 = - 5 + 25 i$

So:

$\left(3 - i\right) \left(1 + 2 i\right) \left(2 + 3 i\right) = - 5 + 25 i$