To identify Jacob's first error in factoring \(5x^2 - 11x + 2\), we need to analyze each step he took.
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Step 1: \(5x^2 + (-10 - x) + 2\)
- This step represents trying to break down the middle term \(-11x\) into two parts. Jacob rewrote \(-11x\) as \(-10 - x\). However, this is incorrect because \(-10 - x\) simplifies to \(-10 - 1x\), which equals \(-10x - x = -11x\). So, this step is actually acceptable in terms of rephrasing the middle term, but it doesn't lead towards the correct factoring.
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Step 2: \((5x^2 - 10x) + (x - 2)\)
- In this step, Jacob separated the expression into two groups: \(5x^2 - 10x\) and \(x - 2\). While \(5x^2 - 10x\) is correct, the grouping of the second part is uncertain because the term needs to reflect the rephrasing from Step 1. The latter group does not accurately connect back to \(-11x\).
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Step 3: \(5x(x - 2) + 1(x - 2)\)
- Here he factored out common terms from the first and second groups identified in Step 2. Again, there’s inconsistently existing which causes problems, especially with \(1\) factored out in this context.
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Step 4: \((5x + 1)(x - 2)\)
- This step represents combining the factors. However, it combines improperly because if the previous step was flawed, this would be inaccurate too.
By breaking down Jacob’s process:
Jacob's first tangible error exists in Step 2 where he improperly categorized the terms of the original polynomial. He stated \(x - 2\) incorrectly; it should remain consistent with representing \(-11x\), which includes \(-10x - x\) instead of separating incorrectly.
Conclusively, while mistakes appear throughout, the first real algebraic error that generates confusion originates in Step 2.
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