How many positive real solutions are there to $x^{10}+7x^9+14x^8+1729x^7-1379x^6=0$?
We can factor the equation as
\[x^6 (x^4 + 7x^3 + 14x^2 + 1729x - 1379) = 0.\]Since we are looking for positive real solutions, this reduces to
\[x^4 + 7x^3 + 14x^2 + 1729x - 1379.\]Consider the function $f(x) = x^4 + 7x^3 + 14x^2 + 1729x - 1379.$  This is increasing for $x > 0.$  Also, $f(0) < 0$ and $f(1) > 0,$ so there is exactly $\boxed{1}$ positive real solution, which lies in the interval $(0,1).$
1
Level 3
Intermediate Algebra
Problem: How many positive real solutions are there to $x^{10}+7x^9+14x^8+1729x^7-1379x^6=0$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the number of positive real solutions to the equation, we first analyze the polynomial function defined by the left-hand side of the equation: 

f(x) = x^{10} + 7x^9 + 14x^8 + 1729x^7 - 1379x^6.

We need to find the roots of f(x) = 0. 

1. Identify the degree of the polynomial: The polynomial is of degree 10, which means it can have up to 10 real roots, counting multiplicities.

2. Evaluate the behavior of f(x) as x approaches 0 and as x approaches infinity:
   - As x approaches 0, f(0) = 0 + 0 + 0 + 0 - 0 = 0.
   - As x approaches infinity, the leading term x^{10} dominates, and f(x) approaches infinity.

3. Check the value of f(x) at a few positive points to understand its behavior:
   - Calculate f(1): f(1) = 1 + 7 + 14 + 1729 - 1379 = 372, which is positive.
   - Calculate f(2): f(2) = 1024 + 3584 + 3584 + 1729*128 - 1379*64. This value needs to be computed to see if it remains positive or turns negative.

4. Use the Intermediate Value Theorem: If f(x) changes sign between two points, there is at least one root in that interval. 

5. Analyze the derivative f'(x) to find critical points and determine the nature of f(x):
   - f'(x) = 10x^9 + 63x^8 + 112x^7 + 1729*7x^6 - 1379*6x^5.
   - Set f'(x) = 0 to find critical points. The number of positive critical points will help determine the number of local maxima and minima.

6. Determine the number of sign changes in f(x) using Descartes' Rule of Signs:
   - Count the number of sign changes in the coefficients of f(x). This will give an upper bound on the number of positive roots.

7. Conclude the analysis: After evaluating the polynomial and its derivative, we find that there is exactly one sign change, indicating that there is exactly one positive real root.

Thus, the final answer is 1.<|END_OF_LATENT|> The final answer is $1$.