00:03
So on a, we have x squared plus x plus 1, and d of x is x minus 1, which means if we set x minus 1 equal to 0, 1 is what's going to go on the outside of our synthetic division, and all the leading coefficients, 1, 1, and 1 are what is going to go on the inside of our synthetic division.
00:33
Let me make my synthetic division a little smaller.
00:35
Bring down the 1.
00:37
1 times 1 is 1.
00:38
1 plus 1 is 2, 1 times 2 is 2, 1 plus 2 is 3.
00:46
Then this 3 serves as your remainder.
00:50
This is your r of x.
00:52
This 2 is your constant term.
00:55
This 1 is the coefficient for x.
00:56
This serves as your, this x plus 2 or, yeah, the x plus 2 serves as your q of x.
01:06
So you end up with, they want us to do p of x.
01:12
Is your d of x times your q of x plus your r of x.
01:23
So our p of x is equal to our d of x was x minus 1.
01:29
Our q of x is x plus 2.
01:32
And our r of x is just 3.
01:34
So if you want to double check to see if you did your math right, and i'm not going to do this on every problem, but i'll just show you on this one.
01:41
This is x times x is x squared plus 2x.
01:47
Inside is minus 1x, minus 2 plus 3, that gives us x squared plus x plus 1.
01:57
And that's what our original one was.
01:59
So you can always double check to see if you did your correct formula, oops, not plus, minus.
02:08
If you did your math right when you did all your synthetic division, and if you put all of your d of x, q of x, and r of x in the right place.
02:17
You can multiply it back out and double check that you got your original.
02:21
So that's a.
02:23
B, they gave us 4x cubed minus x plus 4, and they said our d of x was x minus 2, which means x minus 2 equals 0.
02:42
Our x is going to be 2.
02:43
And notice that you're missing an x squared.
02:46
So we need four.
02:48
0, negative 1, and 4.
02:53
So we're going to use our 2, and our 4 are 0, our negative 1, and 4.
02:59
So 4, 0, negative 1, 4.
03:04
So bring down my 4.
03:06
2 times 4 is 8.
03:09
So you're going to multiply all of these numbers together, and then put them underneath the next number.
03:16
0 plus 8 is 8.
03:18
2 times 8 is 16, negative 1 plus 6.
03:22
Is 15, 2 times 15 is 30, 4 plus 30 is 34.
03:32
This is our remainder, and then what's left here is a 4x squared plus 8x plus 15, and that's our remainder.
03:43
So our p of x ends up being our d, which was x minus 2, times this little guy, and then plus our little remainder that's left.
04:07
That's what you need to write it as.
04:10
So on c, the problem we have with c is that you've got x to the 6th minus 3x to the 5th plus x to the 4th plus 2x cubed, plus x squared minus x plus 1, and the d of x is x squared plus 2.
04:37
So this is not a linear.
04:39
X squared.
04:39
Minus 2 is not linear.
04:43
So you cannot use synthetic division unless you set it equal to zero and get linear.
04:47
Otherwise, you're going to have imaginaries and then it's super hard to do division with imaginaries and it gets really messy.
04:53
So instead, we're going to use long division to do this.
05:01
So we're going to do out long division, x squared plus 2 into x to the 6 minus 3x to the 5th.
05:11
I'm going to do this slowly, plus x to the fourth, plus two x cubed, plus x squared minus x plus 1.
05:25
So my goal is to make x squared match x to the 6th.
05:31
So that's going to be x to the 4th.
05:34
So now i need to multiply these two together.
05:40
X to the 4th times x squared is x to the 6th.
05:43
X to the fourth times two is two x to the fourth so then you need to subtract both of them that's going to give you because you always do subtract that's going to make this disappear bring down my negative 3x to the 5th that's going to mean negative 1x to the 4th i'm going to bring down my next term and i'm going to go back and i'm going to make now my x squared match my negative 3x to the 5th so to make that match, hold on, i'm going to have to go, i have to keep moving the screen back and forth because i need to make it match.
06:28
So let's make this.
06:29
Oh, nope.
06:33
Hold on.
06:36
Okay.
06:36
So then i need a negative 3x cubed to make it match.
06:41
I want to make it match exactly.
06:43
Negative 3x to the 5th and then negative 3x cubed times 2 is negative 6x cubed.
06:50
And again, i need to make, i need to subtract.
06:56
So really i'm going to change the sign for both of these because i have to subtract the whole row.
07:02
So by subtracting, you're just going to change the signs.
07:04
That's going to give me negative 1x to the fourth.
07:10
2x squared or sorry, 2x cubed plus 6x cubed is going to give me plus 8x cubed.
07:19
Bring down my next term plus x squared.
07:24
Then i need to make my x squared match my negative x to the fourth.
07:29
So i need a negative x squared.
07:33
So let me move my screen back up.
07:37
So i need a negative, negative x squared.
07:43
So now i'm going to multiply negative x squared times x squared.
07:47
That's negative x to the fourth.
07:52
And then negative x squared times 2 is negative 2x squared.
07:59
And then again, just like before, i'm going to change my signs, make a match.
08:06
That's going to disappear, just like this one did.
08:10
Bring this down, 8x cubed plus 3x squared.
08:16
I need to go back and bring down, bring down my negative x.
08:28
And now i need to make my 8x cubed match my x squared, which means i need a plus 8x, and then 8x times.
08:44
Times 8x squared is 8x cubed plus 16, oops, plus 16x.
08:58
Again, change the signs.
09:01
Now that's negative...