Which expression is equivalent to \((3\mathrm{y}^2 - 5\mathrm{y} + 2) - (\mathrm{y}^2 - 4\mathrm{y} - 7)\)?2y^2 + y + 92y^2...

1. INFER the required approach

• When subtracting polynomials, we must distribute the negative sign to every term in the second polynomial
• This transforms subtraction into addition of opposite terms
• Strategy: Rewrite as addition, then combine like terms

2. SIMPLIFY by distributing the negative sign

Starting with: \((3\mathrm{y}^2 - 5\mathrm{y} + 2) - (\mathrm{y}^2 - 4\mathrm{y} - 7)\)

Distribute the negative to each term in the second polynomial:

• \(-\mathrm{y}^2\) becomes \(-\mathrm{y}^2\)
• \(-(-4\mathrm{y})\) becomes \(+4\mathrm{y}\)
• \(-(-7)\) becomes \(+7\)

Result: \(3\mathrm{y}^2 - 5\mathrm{y} + 2 - \mathrm{y}^2 + 4\mathrm{y} + 7\)

3. SIMPLIFY by grouping and combining like terms

Group terms with the same variable and power:

• y² terms: \((3\mathrm{y}^2 - \mathrm{y}^2)\)
• y terms: \((-5\mathrm{y} + 4\mathrm{y})\)
• Constant terms: \((2 + 7)\)

Combine the coefficients:

• \(3\mathrm{y}^2 - \mathrm{y}^2 = 2\mathrm{y}^2\)
• \(-5\mathrm{y} + 4\mathrm{y} = -\mathrm{y}\)
• \(2 + 7 = 9\)

Answer: \(2\mathrm{y}^2 - \mathrm{y} + 9\) (Choice C)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when distributing the negative sign, particularly with the middle term of the second polynomial.

For example, they might write: \(3\mathrm{y}^2 - 5\mathrm{y} + 2 - \mathrm{y}^2 - 4\mathrm{y} + 7\) (forgetting to change \(-4\mathrm{y}\) to \(+4\mathrm{y}\))

This gives them: \(2\mathrm{y}^2 + (-5\mathrm{y} - 4\mathrm{y}) + 9 = 2\mathrm{y}^2 - 9\mathrm{y} + 9\), which doesn't match any answer choice, leading to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY reasoning: Students correctly distribute the negative sign but make arithmetic errors when combining like terms.

For instance, they might incorrectly calculate \(-5\mathrm{y} + 4\mathrm{y} = +\mathrm{y}\) instead of \(-\mathrm{y}\), leading them to select Choice A (\(2\mathrm{y}^2 + \mathrm{y} + 9\)).

The Bottom Line:

This problem tests careful execution of the distributive property with negative signs. Success requires both strategic thinking about what subtraction means for polynomials and meticulous attention to signs during algebraic manipulation.