By the polynomial remainder theorem,
will be a factor of
if the remainder upon division is 0, and this remainder is given by
:
Since
, it follows that
. But notice that if
, then we have
and since
, the equation above requires that either
or
, which is impossible. So
.
So we have 8 cases to check:
(1) Notice that if
, we have
. This is only possible for
.
(2) If
, then
, and so we can have
.
(3) If
, then
, and so
.
(4) If
, then
.
(5) If
, then
.
(6) If
, then
.
(7) If
, then
.
(8) If
, then
.
At the same time, we have 8 cases to consider to find how many options there are for
.
(1)
. We have only one choice of
.
(2)
. This is the same as when
, which we found to be 2 choices.
(3) Same as
; 3 choices.
(4) Same as
; 4 choices.
(5) 5.
(6) 4.
(7) 3.
(8) 2.
In total, there are
ways to choose
such that
is a factor of
, so the answer is B.
Note the symmetry of the sum above. You can easily give a slightly briefer combinatorial argument for this answer, but I figured a more brute-force approach would be easier to follow.
It would be 1.25 per pound
Answer:
<h2>a) 0.5367feet</h2><h2>b) 0.5223feet</h2><h2>c) 0.7292feet</h2>
Step-by-step explanation:
Given the rate at which an eucalyptus tree will grow modelled by the equation 0.5+6/(t+4)³ feet per year, where t is the time (in years).
The amount of growth can be gotten by integrating the given rate equation as shown;
a) The number of feet that the tree will grow in the second year can be gotten by taking the limit of the integral from t =1 to t = 2
b) The number of feet that the tree will grow in the third year can be gotten by taking the limit of the integral from t =2 to t = 3
c) The total number of feet grown during the second year can be gotten by substituting the value of limit from t = 0 to t = 2 into the equation as shown
Answer:
70
Step-by-step explanation:
all the angles of a triangle sum 180, so 180 - (70 + 40) = 70
Answer:
The point ( 0,-1) is a solution.
Step-by-step explanation:
y<= (x+1) (x+5)
y>= (x-1)^2 -6
See attached graph.
To prove algebraically, put the point (0,-1) into both equations and see if it is true for both of them.
y<= (x+1) (x+5)
-1 <= (0+1) (0+5)
-1<= 1*5
-1 <=5
True
y>= (x-1)^2 -6
-1 >= (-1)^2 -6
-1 >= 1-6
-1 >=-5
True
The point is a solution