00:01
Okay, suppose i have a polynomial p of x equals x the fourth minus 81.
00:07
And i want to factor this completely using real or complex numbers.
00:11
Recall that complex numbers.
00:13
Everything's a complex number, but it has an imaginary i in it.
00:17
Okay.
00:18
So how would i factor this? let's factor this completely first.
00:22
What i'm going to use is my difference of squares rule, right? something squared minus something else squared, right? this is going to be a plus b times a minus b.
00:30
This is a difference of squares rule.
00:34
So what times itself gives me 81? well, nine times nine gives me 81, and what times itself gives me x to the fourth? well, x squared, right, times x squared, add those exponents, gives me x to the fourth.
00:50
So my a and my b are x squared and nine, so this is just x squared minus nine times x squared plus nine.
00:57
Then i have another difference of squares here, right? what squared gives me x squared will x squared and what squared gives me nine what times itself gives me nine three so this factors as x minus three times x plus three and what i could do here is rewrite this as x squared minus negative nine now what squared gives me x squared x squared and what squared gives me negative nine well, this is going to be 3i, right? i is the same thing as a square of negative 1...