find the positive roots of the equation
2x^3-5x^2+6=0
and evaluate the function y=500x^6 at each root. round you answers to the nearest integer, but only in the final step.
To find the positive roots of the equation 2x^3 - 5x^2 + 6 = 0, we can use the technique called "factorization".
Step 1: Set the equation equal to zero: 2x^3 - 5x^2 + 6 = 0.
Step 2: Look for common factors that can be factored out from each term. In this equation, there are no common factors.
Step 3: Since the equation is a cubic equation, there might be either one real root or three real roots. We can try to find the rational roots by using the Rational Root Theorem.
The Rational Root Theorem states that if a rational number p/q is a root of a polynomial equation, then p must be a factor of the constant term (in this case, 6), and q must be a factor of the leading coefficient (2).
The factors of 6 are ±1, ±2, ±3, and ±6, and the factors of 2 are ±1 and ±2.
We can start by testing the positive factors of 6 divided by the positive factors of 2:
- When x = 1, 2x^3 - 5x^2 + 6 = 2(1)^3 - 5(1)^2 + 6 = 2 - 5 + 6 = 3 (not equal to 0).
- When x = 2, 2x^3 - 5x^2 + 6 = 2(2)^3 - 5(2)^2 + 6 = 16 - 20 + 6 = 2 (not equal to 0).
- When x = 3, 2x^3 - 5x^2 + 6 = 2(3)^3 - 5(3)^2 + 6 = 54 - 45 + 6 = 15 (not equal to 0).
- When x = 6, 2x^3 - 5x^2 + 6 = 2(6)^3 - 5(6)^2 + 6 = 432 - 180 + 6 = 258 (not equal to 0).
Since none of the tested values are roots, we need to use another method to find the positive roots.
Step 4: Graph the equation. Using graphing software or an online graphing tool, plot the graph of the function y = 2x^3 - 5x^2 + 6. From the graph, we can visually determine the positive roots.
Alternatively,
Step 4 (alternative): Use a numerical root-finding method such as the Newton-Raphson method or the bisection method to approximate the positive roots. These methods involve iteration and can provide an approximation for the positive roots.
Once you have found the positive roots (approximations) of the equation, you can evaluate the function y = 500x^6 at each root. Round the results to the nearest integer.
For example, if one of the positive roots is x = 1.5, you can substitute it into the function:
y = 500(1.5)^6 = 500(1.5 × 1.5 × 1.5 × 1.5 × 1.5 × 1.5) = 500(14.0625) = 7031.25 (approx.)
Rounding to the nearest integer, the result is 7031.
Repeat this process for each positive root you found to evaluate y = 500x^6 at each root.