$$ \begin{array}{l}\text { Factor the expression by grouping. } \ 3 x^{3}-2 x^{2}-21 x+14 \ \text { Select the correct choice below and fill in any answer boxes within your choice. } \ \text { A. } 3 x^{3}-2 x^{2}-21 x+14= \ \text { B. The polynomial is prime. }\end{array} $$

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UpStudy AI · Accepted Answer

We start with the expression
$$
3x^3 - 2x^2 - 21x + 14.
$$
We group the terms that have common factors:
$$
(3x^3 - 2x^2) + (-21x + 14).
$$

**Step 1. Factor each group:**

For the first group,
$$
3x^3 - 2x^2 = x^2(3x - 2).
$$

For the second group,
$$
-21x + 14 = -7(3x - 2).
$$

**Step 2. Factor out the common binomial $$(3x-2)$$:**
$$
x^2(3x-2) - 7(3x-2) = (3x-2)(x^2-7).
$$

Thus, the factorization of $$3x^3-2x^2-21x+14$$ is
$$
(3x-2)(x^2-7).
$$
The expression $$3x^3 - 2x^2 - 21x + 14$$ factors to $$(3x - 2)(x^2 - 7)$$.

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