Final answer:
The correct simplified expression for (k-2)/(k²+19k-42) is 1/(k+21), obtained by factoring the denominator and canceling out the common factor of (k-2).
Step-by-step explanation:
When you are given an algebraic fraction like (k-2)/(k²+19k-42), you are often asked to simplify it by factoring the numerator and the denominator and canceling any common factors. Let's start with factoring the denominator because the numerator is already a simple linear expression and cannot be factored any further.
To factor k²+19k-42, we look for two numbers that multiply to -42 and add up to +19. These two numbers are +21 and -2 because 21 * -2 = -42 and 21 + (-2) = 19. So, we can rewrite the denominator as (k+21)(k-2).
Now, the original expression can be written as (k-2)/((k+21)(k-2)). We see that (k-2) is common in both the numerator and the denominator; therefore, it can be canceled out. This leaves us with the simplified expression 1/(k+21).