Factor.3x^2 - 4x

Transcript

00:01 All right, let's do this fun problem together.

00:03 So we have a polynomial that's x to the fifth minus 3x cubed plus 4x squared plus 2x minus 4.

00:08 And the question tells us that one of the roots is 1 minus i.

00:13 Okay, so this is a complex root.

00:15 Complex just means that we have imaginary numbers, i is an imaginary number.

00:18 Okay, so the first thing that we need to realize is that if a polynomial has a complex root in the form a plus bi, which we do have, then the conjugate, which is a minus bi, is also going to be a root.

00:30 Therefore, this is a root and i, sorry, and 1 plus i is also going to be a root.

00:39 So i did this backwards.

00:40 So 1 plus i was the root that they gave us.

00:42 So the root that also has to exist is 1 minus i.

00:46 Okay, so we want to figure out the other three factors.

00:49 And we know that there are five factors in total because the highest index of this polynomial is 5.

00:53 Okay, so when we write things out in factored form, we write it in the form of x minus the root.

00:58 So let's do that with our complex root.

01:00 So we're going to have x minus 1 plus i and then x minus 1 minus i.

01:09 And we're going to have three other roots that we'll solve for later.

01:13 So in order to get to actually solve for those roots, we need to multiply these imaginary roots together.

01:20 Okay, and we're going to use something called the foil method, which is first, outer, inner, last.

01:24 The first thing that i'm going to do is actually distribute this negative into our values, just so things are a little bit simpler for me.

01:31 I don't really like negative numbers, and i think they can sometimes confuse things.

01:35 So let's distribute that.

01:36 So then we're going to be left with x plus negative 1 minus i times x plus negative 1 plus i.

01:50 Okay, we're going to multiply these together again using the foil method.

01:52 So foil, the first term is first.

01:54 So x times x, which is x squared.

01:58 Then we have outer.

02:00 So x times this number here, which is going to be minus x plus x i.

02:10 And then i is inner.

02:12 So this times this, that's going to be minus x minus x i.

02:19 And then l is for last, which is these two numbers here.

02:24 And this is going to be like another mini foil.

02:27 So we have negative 1 minus i times negative 1 plus i.

02:37 Okay, so first negative 1 times negative 1 is 1.

02:42 Outer, negative 1 times i is negative i.

02:45 Inner, negative 1 times negative i is positive i.

02:49 And then last, negative i times i is negative i squared.

02:53 We can actually simplify i squared to be negative 1.

02:56 So we have negative negative 1, which is going to give us positive 1.

02:59 So we have 1 minus i plus i plus 1.

03:03 So this is, these i's will cancel out.

03:06 So we're going to be left here with 2.

03:07 So plus 2.

03:09 Okay, so let's do some cancelling here.

03:11 And i actually forgot to write x squared here.

03:13 This here should be x squared.

03:15 So the positive x i and the negative x i is going to cancel out.

03:20 So we're going to be left with x squared minus 2x plus 2.

03:26 Okay, hopefully that all made sense.

03:28 So we just did a bunch of foiling, okay? so now that we have this polynomial, we're going to divide our big polynomial by this number to get a smaller polynomial that we can then factor to get our remaining three roots.

03:39 So let me rewrite our big polynomial.

03:41 So it's x to the fifth minus 3x cubed plus 4x squared plus 2x minus 4.

03:52 And we're going to do some polynomial long division.

03:54 X squared minus 2x plus 2.

03:57 And i forgot to do something.

03:58 When you're doing polynomial long division, you can't skip an index.

04:02 So notice how we go right from the fifth power to the third power.

04:05 We actually have to erase what we wrote here and have a placeholder for x to the fourth power.

04:11 So minus 3x cubed plus 4x squared plus 2x minus 4.

04:22 Okay, all righty.

04:25 So let's keep going.

04:27 So we're going to start with how many times can x we go into x to the fifth? that's going to be x cubed times.