Let polynomial P be defined as P = -2y^2 + 3y - 9 and polynomial Q be defined as Q...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{P = -2y^2 + 3y - 9}\)
  • \(\mathrm{Q = 4y^2 - 5y + 1}\)
  • Need to find polynomial X such that adding X to P gives Q

• This translates to the equation: \(\mathrm{P + X = Q}\)

2. INFER the solution approach

• Since we have \(\mathrm{P + X = Q}\), we can solve for X by rearranging
• This gives us: \(\mathrm{X = Q - P}\)
• We need to subtract polynomial P from polynomial Q

3. SIMPLIFY through polynomial subtraction

• Set up the subtraction: \(\mathrm{X = Q - P}\)
• Substitute: \(\mathrm{X = (4y^2 - 5y + 1) - (-2y^2 + 3y - 9)}\)
• Distribute the negative sign to each term in P:
  \(\mathrm{X = 4y^2 - 5y + 1 + 2y^2 - 3y + 9}\)

4. SIMPLIFY by combining like terms

• Group similar terms together:
  • y² terms: \(\mathrm{4y^2 + 2y^2 = 6y^2}\)
  • y terms: \(\mathrm{-5y + (-3y) = -8y}\)
  • constant terms: \(\mathrm{1 + 9 = 10}\)

• Final result: \(\mathrm{X = 6y^2 - 8y + 10}\)

Answer: C (\(\mathrm{6y^2 - 8y + 10}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign errors when distributing the negative during polynomial subtraction

Students often write: \(\mathrm{X = (4y^2 - 5y + 1) - (-2y^2 + 3y - 9)}\) but then incorrectly get:
\(\mathrm{X = 4y^2 - 5y + 1 - 2y^2 + 3y - 9}\)

They forget that subtracting a negative makes it positive, so \(\mathrm{-(-2y^2)}\) should become \(\mathrm{+2y^2}\). This leads to wrong coefficients and they might select Choice A (\(\mathrm{-6y^2 + 8y - 10}\)) or Choice B (\(\mathrm{2y^2 - 8y - 8}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Misunderstanding what the problem is asking for

Some students might think they need to add P and Q together instead of finding what to add to P to get Q. This conceptual confusion about the setup leads to random guessing among the answer choices.

The Bottom Line:

This problem tests whether students can correctly set up a polynomial equation from words and execute polynomial subtraction without sign errors. The key insight is recognizing that "what must be added to P to get Q" means finding \(\mathrm{X = Q - P}\).