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Factor the following expression completely.Answer:
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Q. Factor the following expression completely.Answer:
- Rearrange and Combine Terms: First, let's rearrange the terms of the expression in descending order of the powers of . Now, combine like terms.
- Group and Factor: Next, we look for common factors or patterns that might help us factor the expression. We can try to group terms to see if that helps.Group the terms as follows: + +
- Factor by Grouping: Now, let's factor by grouping. We look for common factors in each group.For the first group , we can factor out an , giving us .For the second group , there is no common factor with the other groups, so we leave it as is.For the third group , we can factor out a , giving us .Now we have:
- Check for Mistakes: We notice that there is no common factor between the groups, and the expression does not factor nicely. At this point, we should check if we made any mistakes in the previous steps or if there is a special factoring technique that applies.
- Explore Special Techniques: We can try to factor by grouping in a different way or look for a special product such as a difference of squares, but none of these methods seem to apply here. It's possible that the expression is prime (not factorable) over the integers.
- Use Rational Root Theorem: Since the expression does not factor easily, we can attempt to use the rational root theorem or synthetic division to find any rational roots. However, this is a complex and time-consuming process that may not yield factors. Given the complexity of the expression and the coefficients involved, it is likely that the expression does not have rational factors.
- Conclusion: After attempting various factoring techniques and finding no factors, we conclude that the expression is prime and cannot be factored over the integers.
