Question
Factor by grouping. \[ 6 w^{3}+21 w^{2}-2 w-7 \]
Ask by Barrett Bowman. in the United States
Mar 14,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The expression \(6w^{3}+21w^{2}-2w-7\) factors to \((2w+7)(3w^{2}-1)\).
Solution
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The Deep Dive
To factor the polynomial \( 6 w^{3}+21 w^{2}-2 w-7 \) by grouping, first, we can group the terms as follows: 1. Group the first two terms and the last two terms: \[ (6 w^{3} + 21 w^{2}) + (-2 w - 7) \] 2. Factor out the greatest common factor from each group: \[ 3w^{2}(2w + 7) - 1(2w + 7) \] 3. Now, both groups contain the common factor \( (2w + 7) \): \[ (3w^{2} - 1)(2w + 7) \] So, the factored form of the polynomial is: \[ (3w^{2} - 1)(2w + 7) \]
