MTH1010_05_03_2

Question

MTH1010_05_03_2

BlackTom AI · Accepted Answer

The question shown asks for the third zero x = _____ for the cubic polynomial depicted. However, the provided data includes no answer choices to evaluate, so I will outline a clear method to find the third zero and then apply it to the polynomial mentioned in the answer text.

First, restating what’s given: the answer block claims the zeros are x = -2, 0, and 3, and that the third zero should be determined. There is a mismatch between the text in the image and the explicit answer provided, so we’ll focus on the polynomial form described in the answer field to resolve the third zero.

Step 1: Write the polynomial and factor if possible. The stated polynomial in the analysis text is f(x) = x^3 − x^2 − 6x. Factor out the greatest common factor first:
f(x) = x(x^2 − x − 6).

Step 2: Factor the quadratic. The quadratic x^2 − x − 6 factors as (x − 3)(x + 2) because the numbers −3 and +2 multiply to −6 and add to −1:
x^2 − x − 6 = (x − 3)(x + 2).

Step 3: Write the full factorization. Therefore:
f(x) = x(x − 3)(x + 2).

Step 4: Find all zeros. Setting f(x) = 0 gives the zeros:
x = 0, x = 3, x = −2.
This confirms the three x-intercepts (zeros) are −2, 0, and 3.

Step 5: Identify the third zero. If you’re asked for the third zero after two zeros have been identified, the remaining zero is the one not yet listed. Depending on the order, the set of zeros is {−2, 0, 3}, so the third zero could be either −2, 0, or 3 if you are enumerating in a particular sequence. In a standard ascending order, the zeros are −2, 0, 3, so the third zero is 3.

Common potential confusions to avoid:
- If you attempt to guess based on a graph, ensure you read the intercepts correctly and that the polynomial is exactly the one given; a different constant term or coefficient would change the zeros.
- Do not assume there are only two zeros for a cubic; a cubic with real coefficients can have three real zeros (as here) or one real and two complex zeros.

Summary from the factoring approach: the zeros are −2, 0, and 3; hence the third zero, depending on the order you present them, is 3 if you list in ascending order, or simply the remaining distinct zero not yet named.

MCD2130 - T2 - 2025 6.8 Online Quiz MCD2130_QUIZ_FIVE Polynomials and Rational Functions

MTH1010_05_03_2

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