Final answer:
To find the years in which the company's profit was zero, we first try to factor the quartic profit function P(t). If it's complex, numerical methods or graphing might be needed. For the separate quadratic equation t² + 10t - 2000 = 0, the quadratic formula is used to find the solutions for t.
Step-by-step explanation:
The question requires us to factor the profit function P(t) = t⁴ - 10t² + 9 to find the years in which the profit was zero. To find the values of t, we must identify the values for which P(t) equals zero.
The given function is a quartic equation, and we can attempt to factor it by grouping terms or other factoring techniques that apply to polynomial equations.
If it does not factor easily, we may need to use numerical methods or graphing to estimate the values for t. However, the question later presents a quadratic equation, t² + 10t - 2000 = 0, which is presumably part of the same problem but somehow incorrectly included. To solve this quadratic equation for t, we employ the quadratic formula:
t = (-b ± √(b²-4ac))/(2a), where a = 1, b = 10, and c = -2000.
Inserting these values into the quadratic formula, we can solve for t. Remember, this function models profit in thousands of dollars, so the solution will tell us in which years since 2000 the company had zero profit, in terms of thousands of dollars.