Question

Error Analysis: In order to identify the zeros of the function, a student factored the cubic function f(x) = x³ - 3x² - 10x as follows:

a. x(x - 3)(x + 10)
b. (x - 5)(x + 2)(x + 1)
c. x(x + 2)(x - 5)
d. (x + 5)(x - 2)(x - 1)

Answer 1

The cubic function f(x) = x³ - 3x² - 10x is correctly factored as option c: x(x + 2)(x - 5), yielding zeros of 0, -2, and 5. The other options do not represent the correct factorization when expanded.

Step-by-step explanation:

Identifying the zeros of a cubic function is a standard task in algebra that involves factoring. Specifically, the cubic function f(x) = x³ - 3x² - 10x can be factored to find its zeros, assuming the factoring is done correctly. The options provided require evaluation to determine the correct factorization.

Looking at the options:

• Option a: x(x - 3)(x + 10) does not produce the original cubic function when multiplied out.
• Option b: (x - 5)(x + 2)(x + 1) also does not give the original function when expanded.
• Option c: x(x + 2)(x - 5), which simplifies to x³ - 5x² + 2x² - 10x = x³ - 3x² - 10x, correctly represents the original function.
• Option d: (x + 5)(x - 2)(x - 1) is not the correct factorization either.

Thus, the correct factorization is option c: x(x + 2)(x - 5), revealing the zeros of the function as 0, -2, and 5. The mistake made with the other factorizations relates to incorrect combinations of numbers that, when multiplied, should give the coefficients and constant term of the original cubic function.