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\( 1 \leftarrow \) 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.

Ask by Gibson Bartlett. in the United States
Feb 24,2025

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A. The answer is \( a(2a - 3)(7a - 6) \).

Solución

Step 1. Notice that every term in 14a³ - 33a² + 18a has a common factor a, so factor that out:   a(14a² - 33a + 18) Step 2. Now, factor the quadratic 14a² - 33a + 18. To do this, we want to find two numbers that multiply to (14 × 18) = 252 and add to -33. The numbers -21 and -12 work since (-21) × (-12) = 252 and (-21) + (-12) = -33. Step 3. Rewrite -33a using -21a and -12a:   14a² - 21a - 12a + 18 Step 4. Factor by grouping:   Group the terms: (14a² - 21a) - (12a - 18)   Factor out common factors in each group:     7a(2a - 3) - 6(2a - 3) Step 5. Notice that (2a - 3) is common, so factor it out:   (2a - 3)(7a - 6) Step 6. Combine back with the factor a from step 1:   a(2a - 3)(7a - 6) Thus, the completely factored form of the trinomial is: A. The answer is a(2a - 3)(7a - 6).

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Bonus Knowledge

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 \( a(7a - 9)(a - 1) \).

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