# Video: Finding the Product of Two Complex Numbers in Algebraic Form

Multiply (β3 + π) by (2 + 5π).

02:01

### Video Transcript

Multiply negative three plus π by two plus five π.

There are a couple of different things we need to recall here. Firstly, we see that we have two complex numbers of the form π plus ππ. Remember, π is the real component of the complex number and π is its imaginary component. We also recall that π is the solution to the equation π₯ squared equals negative one such that π is often written as the square root of negative one.

Weβre looking to find the product of negative three plus π and two plus five π. And when we write it like this, we spot that this looks a lot like the product of two binomials. In fact, we deal with this in much the same way. We distribute our parentheses using either the FOIL or grid method. Letβs use the FOIL method.

Remember, βFβ stands for βfirst.β We multiply the first term in the first bracket by the first term in the second. Negative three times two is negative six. βOβ stands for βouter.β We multiply the outer terms in our expressions. Negative three times five is negative 15. So negative three times five π is negative 15π. βIβ then stands for βinner.β We multiply the inner terms. That gives us two π. And then, finally, βLβ stands for βlast.β We multiply the last term in the first bracket by the last term in the second. And that gives us five π squared.

Now weβre not quite done here. We can simplify this further. Firstly, we spot that we can collect some like terms, negative 15π and two π. That gives us negative six minus 13π plus five π squared. But then we recall that π is equal to the square root of negative one. This must mean that we can say that π squared is equal to negative one. We therefore replace π squared with negative one. And we see that the product of these two complex numbers is negative six minus 13π minus five, which is negative 11 minus 13π. The product of the two complex numbers negative three plus π and two plus five π is negative 11 minus 13π.