Final answer:
To write h(x) as a product of linear factors, identify the roots and express each as a factor (x - root). If necessary, use a calculator to find and round the roots. For non-integer exponents, calculators help in finding numerical values rather than factors.
Step-by-step explanation:
To write h(x) as a product of linear factors, you must first identify the roots of the function. Once the roots are identified, each root can contribute to a linear factor of the form (x - root). If tools like a calculator or computer are necessary to find approximate roots, especially when dealing with higher-degree polynomials, make sure to round off the roots to four decimal places as per the instructions.
Consider a polynomial equation h(x) = 80x. Factoring out the greatest common factor, which is 80, we get h(x) = 80(x). Here, 'x' is already a linear factor, thus the product of linear factors for h(x) is simply 80 * (x).
In the case where you have a function that is more complex, such as f(x) with mixed terms like w - / 5.4^2 - 7, you would need to simplify the equation first. If the expression contained real and distinct roots, you would use the quadratic formula or other methods to find them and then express the polynomial as a product of its linear factors.
The example equation -kxdx = -kx^2 demonstrates a differential equation. If you were to solve this, you would integrate both sides to find a function of x that differentiates to the equation given, but this doesn't express the solution as a product of linear factors directly.
To deal with non-integer exponents as in 3^1.7, calculators provide a way to find these values accurately. It's not about writing it as a product of linear factors, but rather about finding the numerical value.
Lastly, understanding the exponent rule (x^a)^b = x^(a.b) is crucial in simplifying expressions before factoring them into linear terms.