00:01
We will now write the polynomial whose graph is shown here.
00:04
So this basically represents a graph of the polynomial p of x.
00:09
And so to write the polymer p of x, we need a few information, which we could get from the graph of the p of x.
00:16
That is, we need the zeros of p of x as plus the y intercept of the p of x, which we can get from the graph.
00:30
When you look at the graph of this polynomial p of x we see that x equal to negative 4 is 1 -0 because the graph intersects the x -axis at negative 4 as well as the graph cuts the x -axis at negative 3 so we have 2 zeros negative 4 and negative 3 and we also see another 0 at x equal to 1 but this one has a multiplicity of 2 because it touches the graph at x equal to 1 and let's also identify the y intercept the graph intersects the y axis at x equal to 0 so we see that the y intercept is 12 and so we have the information that is 0s are negative 4 this is negative 4 negative 3 and 1 with multiplicity of 2 and the y intercept of the graph of p of x is now let's write the polynomial p of x.
01:35
So i write polynomial p of x and this equals, i put this constant a which is a polynomial constant times the factors for each of the zeros.
01:47
That is we write a linear factor for each of the zero.
01:51
So i put x minus of the first factor is minus four and another factor for another zero.
01:58
That is x minus of minus three.
02:01
This is for the 0 minus 3 and these two are multiples of having a multiplicity of 1 so i don't need to put any power to this one or it just consider that they have the power of 1 and then i put another linear factor for the 0 1 so i put x minus 1 but since this has a multiplicity of 2 i write down this as a power of 2 let me simplify this first so we have a p of x and this equals a is the polynomial constant times x minus of minus 4 is x plus 4 times x minus of minus 3 is x plus 3 and then x minus 1 quantity square.
02:50
Now using the y intercept information we can determine the value of a.
02:55
So we have to plug in at x equal to 0, p of x equal to 12.
03:03
Because the y intercept is 12.
03:06
So replace p of x by 12 and also put x equal to 0 in this function or this polynomial.
03:14
So this is a times of replace x by 0 becomes 0 plus 4 times 0 plus 3 times 0 minus 1 quantity is so let's determine a from this one.
03:28
So this gives 12 equals 0 plus 4 is 4.
03:33
0 plus 3 is 3 and then 0 minus 1 is minus 1.
03:38
But minus 1 quantity squared and this equals minus 1 multiplied with itself 2 times.
03:44
So it is 1.
03:45
So this is 1 here.
03:47
And so we have the equation 12 equals 4 times of 3 is 12 and then 12 times of a is 12.
03:55
So this cues a equals, we divide both sides by 12, we get a equal to 1.
04:01
And so we have now determined the value of a.
04:04
We can plug it into the polynomial equation.
04:08
So therefore we get p of x and this equals 1 times of x plus 4 times x plus 3 times x minus 1 quantity squared.
04:26
And so this is the polynomial.
04:29
I can just remove this one because one times of all the linear factors will be the same linear factors.
04:36
But this is in factor form which we can give it in powers of x form.
04:42
So for that we have to multiply each of the terms and then give it in the powers of x form.
04:49
So for that first i'm going to expand this using this binomial.
04:54
That is i can use this binomial identity that is a minus b quantity squared and this equals a squared plus b squared minus of two times of ab and so i can use this for this for the expansion of this binomial that is x minus 1 quantity square consider this x as a and this one as b so therefore this becomes x squared plus b is 1 so it is 1 so it is 1...