Let's analyze Jacob's steps to find out where he made the first error in factoring the quadratic expression \(5x^2 - 11x + 2\).
Step 1:
He rewrites the equation as:
\[ 5x^2 + (-10x - x) + 2 \]
This is correct since \(-10x - x = -11x\).
Step 2:
He rewrites it as:
\[ (5x^2 - 10x) + (x + 2) \]
Here, he incorrectly grouped the terms. He should group it as \[ (5x^2 - 10x) + (-x + 2) \] to maintain the correct expression. This is the first error.
Step 3:
He factors out from the first group:
\[ 5x(x - 2) + 1(x - 2) \]
This part is correct since \((5x^2 - 10x) = 5x(x - 2)\) and \((x - 2) = 1(x - 2)\).
Step 4:
He combines terms to arrive at the final factors:
\[ (5x + 1)(x - 2) \]
This part is mathematically correct based on his grouping but incorrect based on the original polynomial \(5x^2 - 11x + 2\). The original quadratic cannot be factored to this product.
In conclusion, Jacob made his first error in Step 2 when he incorrectly grouped the terms. The correct grouping should have been \((5x^2 - 10x) + (-x + 2)\).
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