The sum of -{2x^2 + x + 31} and 3x^2 + 7x - 8 can be written in the form...
GMAT Advanced Math : (Adv_Math) Questions
The sum of \(-2\mathrm{x}^2 + \mathrm{x} + 31\) and \(3\mathrm{x}^2 + 7\mathrm{x} - 8\) can be written in the form \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\), where \(\mathrm{a}\), \(\mathrm{b}\), and \(\mathrm{c}\) are constants. What is the value of \(\mathrm{a} + \mathrm{b} + \mathrm{c}\)?
1. TRANSLATE the problem information
- Given information:
- First polynomial: \(-2\mathrm{x}^2 + \mathrm{x} + 31\)
- Second polynomial: \(3\mathrm{x}^2 + 7\mathrm{x} - 8\)
- Need to find their sum in the form \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\)
- Then calculate \(\mathrm{a} + \mathrm{b} + \mathrm{c}\)
2. SIMPLIFY by adding the polynomials
- Set up the addition: \((-2\mathrm{x}^2 + \mathrm{x} + 31) + (3\mathrm{x}^2 + 7\mathrm{x} - 8)\)
- Combine like terms:
- \(\mathrm{x}^2\) coefficients: \(-2 + 3 = 1\)
- \(\mathrm{x}\) coefficients: \(1 + 7 = 8\)
- constant terms: \(31 + (-8) = 23\)
- Result: \(\mathrm{x}^2 + 8\mathrm{x} + 23\)
3. TRANSLATE to identify coefficients
- In the form \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\), we have:
- \(\mathrm{a} = 1\) (coefficient of \(\mathrm{x}^2\))
- \(\mathrm{b} = 8\) (coefficient of \(\mathrm{x}\))
- \(\mathrm{c} = 23\) (constant term)
4. SIMPLIFY the final calculation
- \(\mathrm{a} + \mathrm{b} + \mathrm{c} = 1 + 8 + 23 = 32\)
Answer: 32
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make sign errors when combining terms, especially with the constant terms \((31 + (-8))\).
Many students incorrectly calculate \(31 + (-8)\) as \(31 + 8 = 39\) instead of \(31 - 8 = 23\). This gives them a final polynomial of \(\mathrm{x}^2 + 8\mathrm{x} + 39\), leading to \(\mathrm{a} + \mathrm{b} + \mathrm{c} = 1 + 8 + 39 = 48\). This leads to confusion since 48 isn't typically among the answer choices, causing them to second-guess their work.
Second Most Common Error:
Poor TRANSLATE reasoning: Students forget what the problem is actually asking for and stop after finding the sum polynomial.
They correctly determine that the sum is \(\mathrm{x}^2 + 8\mathrm{x} + 23\) but then select this as their final answer instead of calculating \(\mathrm{a} + \mathrm{b} + \mathrm{c} = 32\). This causes them to get stuck when they can't find \(\mathrm{x}^2 + 8\mathrm{x} + 23\) among the numerical answer choices.
The Bottom Line:
This problem tests your ability to carefully perform polynomial addition with attention to signs, then remember to complete the final step of adding the coefficients together.