00:01
Given the function f of x equal 8 x to the 4th minus 14 x cube, minus 71 x squared minus 10 x plus 24, in part a, we will verify that x plus 2 and x minus 4 are factors of this function.
00:22
In part b, we find the remaining factors of f.
00:27
In par c, we write the complete factorization of f, with the t here, par d we list all real zeros of f and in part e we confirm the result by graphing f.
00:50
So in part a we verify that x plus 2 and x minus 4 are factors of f.
00:58
So it means that if for example x plus 2 is factor of f if and only if f at negative 2 is 0.
01:18
That's the same thing because the function is written as x plus 2 times another polynomial.
01:25
And in that case, the factor x plus 2 becomes 0 when x equal negative 2.
01:31
So f at negative 2 got to be 0.
01:36
Each time we have a factor, we have a root and when we have a root, we have a factor.
01:40
That is, it's equivalent.
01:44
So we verified this.
01:46
So we got to take the function and evaluate that at negative 2.
01:50
8 times negative 2 to the 4th minus 14 negative 2 cube minus 71 x that is negative 2 squared minus 10 times negative 2 plus 24 that is f at negative 2 2 and that's equal 2 negative 2 and that's equal to negative 2 to the 4th is positive because the power is even so yet 8 times 16 minus 14 times and in this case negative 2 square is negative 8 because the power is odd then here minus 71 times 4 because the power is even so we get 2 squares 4 plus 20 negative 10 times negative 2 is 20 plus 24 and so so we get here 16 times 8, 128.
03:12
Then we have plus, minus times minus, and 14 times 8 is 112 minus, minus 284 plus and 20 plus 24 is 44.
03:37
44, so we get, now we get 100 ,000.
03:52
128 plus 112 plus 44 and that's it that is in 14 at 1 8 and 2 it's perfect so it's 184 minus 184 that is i have some or add up these three terms here and i got 284 so we have 284 minus 184 minus 294 is 0.
04:33
So we have proof then f and negative 2 is 0 and so x plus 2 is a factor of f.
04:57
Okay, we do the same for x negative x minus 4.
05:01
So x minus 4 is a factor of f if and only if f at 4 in this case is because of value that unifies this factor is x equal 4 so we calculate f at 4 is 8 times 4 to the 4 minus 14 times 4 q and then negative 71 times 4 squared minus 10 times 4 plus 24.
05:50
Now we calculate 4 to the fourth, that is 56 minus 14 times 4 cube is 64.
06:17
Here is 14 minus 71 times 16 minus 40 plus 24 and that is 256 and that is 2506.
06:30
Times 8 is 2048 minus 14 times 64 is 8996 minus 796 minus 71 times 16 is 1136 minus 40 plus 24 and now we add the positives this case is 1 ,000 this one we get here going to do it here 2048 plus 24 is 12 get 1 6 and 1 7 0 2 so it's 2072 we get it here and now we do these 3 terms and we will get minus the sum of these 3 terms 896 1136 and 40 40 and that guess 12 get 1 10 13 17 get 1 10 13 17 get 1 again 10 and 2 that's perfect that's 0 and this is a blue so we get this can get rid of this the operation here.
08:12
So we have that and then we say the same thing about.
08:16
Then f at 4 is 0.
08:22
And so x minus 4 is a factor of f.
08:36
So we have two factors of f.
08:40
Now we go for part b.
08:42
In part b, we want to find the remaining factors.
08:47
And because these are two factors, can do a division if we want by the polynomial the polynomial function of the factors x plus 2 and x minus 2 so we get 8 then we have negative 14 negative 71 negative 10 and 24 these are the coefficients of f in ascending descending order of the powers of x and we have for example here we have uh four which is one of the factors one of the zeros of f so we get eight 32 minus 14 is 18 times 4 72 minus 71 is 1 times 4 72 minus 10 is negative 6 times 4 is negative 24 is 0.
10:16
So we can say that, doing this synthetic division here, we can say that 8x to the 4th minus that is f.
10:33
This is f, first of all, 8x to the 4th minus 14x cubed, minus 71 x square, minus 10 plus 24.
10:49
4 is equal to x minus 4 times 8 and we know the quotient of the division of the polygon given polynomial by x minus 4 is 1 degree less that is cube it's cubic polynomial so it's 8 x cube plus 18 x square plus x or 1x minus 6 and the other route we've found before, which is negative 2, get a bit of this polynomial.
11:31
Because we have only factorized or use a factor x minus 4...