00:09
If p of x is equal to 3x par 5 minus 5 x power 4 minus 8 x cube plus 16 x square plus 21 x plus 5.
00:52
So we have to find all zeros of this polynomial.
00:58
So using rational zero theorem we see that all point.
01:15
Possible zeros are less or minus one, less or minus 1 by 3, less or minus 5 by 3.
01:45
So we'll check these values, the polynomial.
01:50
So first, when x is equal to 1, the polynomial will become 3 minus 5 minus 8, plus 16, plus 21 plus 5 that is not going to be equal to 0.
02:09
Similarly, when x is equal to minus 1, the polynomial will become minus 3 minus 5 plus 8 plus 16 minus 21 plus 5 that is going to be equal to 0.
02:42
So now we'll reduce a polynomial using synthetic division and we know that minus 1 is a 0.
02:50
0.
02:54
So we'll write down the coefficients of the polynomial.
03:02
So that is going to be 3 minus 5 minus 8 plus 16 21 and 5.
03:45
So 3 plus 0 is 3.
03:49
3 into minus 1 minus 3 minus 1 minus 1 minus 1 into minus 8 minus 1 into minus 8 8 minus 1 into minus 8 8 0 0 minus 0 0 0.
04:01
1 into 0, 16 plus 0 16, 16 to minus 1 minus 16.
04:10
21 minus 16 is 5.
04:13
It's going to be minus 5.
04:16
It's going to be minus 0.
04:32
So using the graph, if we see the curve intersect x axis at minus 1 and there is a deviation turning point at this point.
04:51
And we found minus 1 is a 0.
04:53
Now since there is a turning point it is of even multiplicity so minus one will be double will again reduce this polynomial with minus one as zero.
05:35
So three plus zero three three into minus one minus three minus three minus eight minus eleven minus eleven into minus one is eleven.
05:47
11 plus 0 is 11 11 into minus 1 is minus 11 16 minus 11 is 5 5 into minus 1 is minus 5 that is going to be 0 so the result in polynomial will be 3 x q minus 11 x square plus 11 x plus 5 is equal to 0 so using rational zero theorem theorem we found that the possible zeros are less or minus one less or minus one by three there's a minus five less or minus five by three so from the graph we found that minus one is a double and there is another intersecting point between zero and minus 0 .5, it is minus half...