00:01
Hey students.
00:03
So this question is asking us to multiply two taylor series polynomials together.
00:09
So we have e to the x, which i have written out as 1 plus x plus x squared over 2 factorial and so on.
00:16
And sign of x which is equal to x minus x cube divided by 3 factorial plus x to the fifth divided by 5 factorial.
00:26
And let me change colors actually.
00:28
You can see those written out right here.
00:32
And to multiply them together, it's actually pretty straightforward.
00:36
You just multiply the two functions together.
00:40
And so here i have written out all the functions.
00:44
So they want us to find the expansion up to the fifth polynomial.
00:51
So that's why we have it written out to the fifth up here.
00:57
Also, if you know, i just solved the factorials just because we'll be multiplied.
01:03
Them together to reduce fractions later so i just went ahead and solved them two factorial two times one is two three factorial is three times two which is six four factorial just multiply that by four then multiply that by five and so on and you get 24 and 120.
01:21
So solving just multiplying these straight through you're just factoring like always so you multiply one by x and so on so on through and through so it'll take me a while right here.
01:39
This out.
01:40
So sorry, bear with me.
01:41
One times x is x.
01:46
Then what did i do? i did all the x functions first.
01:51
So, okay.
01:53
So then x times x is just x squared.
02:01
And next we have x to the second, x squared, divided by 2 times x, which is just x to the cubed.
02:14
Sorry about that.
02:16
Over 2 plus x to the third.
02:25
Oh, sorry, x to the fourth.
02:29
Went ahead and messed up there.
02:34
So x to the third times x is x to the 4 over 6.
02:41
The first terms are pretty easy for this one.
02:47
X to the 4 times x is x to the 5th over 24.
03:03
And then x to the fifth over 120 it just turns into x to the 6 which actually that's a power that we don't have to do because we just want to find terms to x to the 5th so we could actually go ahead and skip that term and we could move on to the second term in the sign function which is a minus x cubed divided by 6 so that minus 1 obviously or times 1 is just itself so we have a minus x cubed over 6 and you could see where we'll be combining the terms later.
03:39
We have an x cubed here and an x cubed here.
03:45
So now this term times that just turns into a minus x to the fourth over six.
03:50
And you could see that those terms are going to cancel later.
03:58
And we just have to do this term with the next term as well, which is a minus x to the fifth.
04:03
Because again, we just want powers to the fifth.
04:05
So everything else we multiply the x cubed term by will be greater than.
04:10
Power of five so then we don't need to do that and then six times two is 12 so here we go that down there and the only term with this is just itself so we just add in x to 5 120 divided by here and again that's just because uh we could continue to multiply these functions all the way out but x to the fifth times x x x to x to the 6 and you know so on and so on you just get higher powers of x and they only want us to find up to x to the fifth root.
04:58
So this is our entire factored equation.
05:04
And now all you have to do is combine like terms.
05:07
And like i said earlier, let's change color to blue.
05:11
X to the fourth.
05:13
You could see x to the fourth divided by six...