0:00
Hello.
00:01
Okay, so here where we want a polynomial of degree five, where we have zeros at, well, x equals zero and then zero again.
00:08
That means you have a repeated zero at x equals zero, and then negative one, eight, and negative eight.
00:12
So if zero is a zero, that just means that x minus zero or just x is going to be a factor, but it's repeated.
00:19
So we're going to have x squared.
00:20
So we have basically x minus zero squared, which is x squared.
00:23
And then since negative one is a zero, that means that x plus one is a factor.
00:28
And 8 is a 0, meaning that x minus 8 is a factor, and negative 8 is a 0, meaning that x minus negative 8, or x plus 8, is a factor.
00:39
Okay, so now we have something in the form, well, of like this, but then we have some a value, right? so, a times this.
00:46
But if we know that f of 1 is equal to negative 126, we have that y is equal to this, well, that means that if we set x equal to 1, the output is negative 126.
00:57
So that means we have negative 126 is going to be equal to a times, while just one square.
01:03
So that's times one.
01:04
One plus one is two, so times two.
01:08
One minus eight is what, negative seven.
01:11
And then eight, one plus eight is going to be, no, one minus eight is not negative.
01:19
One minus eight, yeah, it's negative seven.
01:21
And then one plus eight is nine.
01:23
So times nine.
01:24
So we're going to have something then of the four.
01:27
Negative 126 is equal to, well, a times one, times two, times negative seven times nine.
01:32
So if we multiply that out, we're going to have what? well, one times two is just two and then times negative seven.
01:42
So that's going to be negative 14 and then times nine, which is 126.
01:47
So this is equal to negative 126.
01:50
So we end up here with negative 126 is equal to a times, well, negative 126.
01:57
Or is equal to negative 126a.
02:01
Well, therefore, a is equal to 1.
02:03
So if a is equal to 1, we basically are polynomial then becomes y is equal to, the value of a is just 1.
02:10
So we have x squared times x plus 1, and then times x minus 8 times x plus 8.
02:22
Now, if we really wanted to, we could distribute this to actually see.
02:26
So we could leave it like so, or if we distribute, so let's go ahead and distribute the x plus 8 times x minus 8 first...