<span>I note that this problem starts out with "Which is a factor of ... " This implies that you were given several answer choices. If that's the case, it's unfortunate that you haven't shared them.
I thought I'd try finding roots of this function using synthetic division. See below:
f(x) = 6x^4 – 21x^3 – 4x^2 + 24x – 35
Please use " ^ " to denote exponentiation. Thanks.
Possible zeros of this poly are factors of 35: plus or minus 1, plus or minus 5, plus or minus 7. Use synthetic division; determine whether or not there is a non-zero remainder in each case. If none of these work, form rational divisors from 35 and 6 and try them: 5/6, 7/6, 1/6, etc.
Provided that you have copied down the function
</span>f(x) = 6x^4 – 21x^3 – 4x^2 + 24x – 35 properly, this approach will eventually turn up 1 or 2 zeros of this poly. Obviously it'd be much easier if you'd check out the possible answers given you with this problem.
By graphing this function, I found that the graph crosses the x-axis at 7/2. There is another root.
Using synth. div. to check whether or not 7/2 is a root:
___________________________
7/2 / 6 -21 -4 24 -35
21 0 -14 35
----------- ------------------------------
6 0 -4 10 0
Because the remainder is zero, 7/2 (or 3.5) is a root of the polynomial. Thus, (x-3.5), or (x-7/2), is a factor.
The normal vectors to the two planes are (3, 3, 2) and (2, -3, 2). The cross product of these will be the direction vector of the line of intersection, (12, -2, -15).
Using x=0, we can find a point on this line by solving the simultaneous equations that remain:
... 3y +2z = -2
... -3y +2z = 2
Adding these, we get
... 4z = 0
... z = 0
so the point we're looking for is (x, y, z) = (0, -2/3, 0). This gives rise to the parametric equations ...
- x = 12t
- y = -2/3 -2t
- z = -15t
By letting t=2/3, we can find a point on the line that has integer coefficients. That will be (x, y, z) = (8, -2, -10).
Then our parametric equations can be written as
- x = 8 +12t
- y = -2 -2t
- z = -10 -15t
1. To solve this exercise you must plot the vertices of the rectangle: (3,9);(5,9); (3,3); (5,3). The graph is shown in the figure attached.
<span>
2. As you can see in the graph, the rectangle has 6 feet long (L=6 feet) and 2 feet wide (W=2 feet).
3. Then, the perimeter of the rectangle is:
P=2L+2W
4. When you substitute the values of "L" and "W" into </span>P=2L+2W<span>, you obtain:
P=2(6 feet)+2(2 feet)
P=12 feet+4 feet
P=16 feet
Therefore, the total length of border needed is 16 feet.</span>
