Answer:
x = -7
Step-by-step explanation:
Our function is . Notice that our possible roots are when x + 4 = 0 and when x + 7 = 0. So, our roots are -4 and -7.
However, the power above x + 4 is even, meaning the graph will simply <em>touch</em> the x-axis at x = -4, but not pass through. The power above x + 7, though, is odd, which means the graph will <em>cross</em> the x-axis.
Thus, the answer is x = -7.
Answer:
Amount invested at 8 % rate = x = $ 15000
Amount invested at 9 % rate = 34000 - x = 34000 - 15000 = $ 19000
Step-by-step explanation:
Total Amount = $ 34000
Let amount invested at 8 % rate = x
Amount invested at 9 % rate = $ 34000 - x
Total interest = $ 2910
291000 = 8 x + 306000 - 9 x
x = 306000 - 291000
x = 15000
So amount invested at 8 % rate = x = $ 15000
Amount invested at 9 % rate = 34000 - x = 34000 - 15000 = $ 19000
When , we have
and of course 3 | 6. ("3 divides 6", in case the notation is unfamiliar.)
Suppose this is true for , that
Now for , we have
so we know the left side is at least divisible by by our assumption.
It remains to show that
which is easily done with Fermat's little theorem. It says
where is prime and
is any integer. Then for any positive integer
,
Furthermore,
which goes all the way down to
So, we find that
QED