\( 1 \leftarrow \) Factor the trinomial. \( 14 a^{3}-33 a^{2}+18 \mathrm{a} \) Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The answer is \( \square \). (Factor completely.) B. The trinomial is not factorable.
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To factor the trinomial \( 14a^{3} - 33a^{2} + 18a \), we first look for a common factor in all the terms. Here, we can factor out \( a \), giving us \( a(14a^{2} - 33a + 18) \). Next, we need to factor the quadratic \( 14a^{2} - 33a + 18 \). Using the AC method, we look for two numbers that multiply to \( 14 \times 18 = 252 \) and add to \( -33 \). Those numbers are \( -18 \) and \( -14 \). Now we rewrite the middle term: \( 14a^{2} - 18a - 14a + 18 \). Grouping gives us \( 2a(7a - 9) - 2(7a - 9) \). Factor by grouping: Thus, the complete factorization is \( a(7a - 9)(2a - 2) \). So, the answer is \( a(7a - 9)(2a - 2) \) or simply \( a(7a - 9)(a - 1) \) after simplifying. So, A. The answer is \( a(7a - 9)(a - 1) \).
