Question: Let \(\mathrm{P(x)}\) be the polynomial obtained when \(\mathrm{(x - 2)(3x + 5)}\) + \(\mathrm{(2x - 1)(x - 4)}\) is...
GMAT Advanced Math : (Adv_Math) Questions
Question:
Let \(\mathrm{P(x)}\) be the polynomial obtained when \(\mathrm{(x - 2)(3x + 5)}\) + \(\mathrm{(2x - 1)(x - 4)}\) is expanded and like terms are combined. If \(\mathrm{P(x)}\) can be written in the form \(\mathrm{ax^2 + bx + c}\), where \(\mathrm{a}\), \(\mathrm{b}\), and \(\mathrm{c}\) are constants, what is the value of \(\mathrm{a + b + c}\)?
1. TRANSLATE the problem information
- Given information:
- \(\mathrm{P(x) = (x - 2)(3x + 5) + (2x - 1)(x - 4)}\)
- Need to find \(\mathrm{a + b + c}\) where \(\mathrm{P(x) = ax^2 + bx + c}\)
2. INFER the solution approach
- We have two main options:
- Method 1: Expand everything and combine like terms
- Method 2: Use the fact that \(\mathrm{P(1) = a + b + c}\) (elegant shortcut!)
- Let's use Method 1 first for practice, then verify with Method 2
3. SIMPLIFY the first binomial product
- \(\mathrm{(x - 2)(3x + 5) = x(3x + 5) - 2(3x + 5)}\)
- \(\mathrm{= 3x^2 + 5x - 6x - 10}\)
- \(\mathrm{= 3x^2 - x - 10}\)
4. SIMPLIFY the second binomial product
- \(\mathrm{(2x - 1)(x - 4) = 2x(x - 4) - 1(x - 4)}\)
- \(\mathrm{= 2x^2 - 8x - x + 4}\)
- \(\mathrm{= 2x^2 - 9x + 4}\)
5. SIMPLIFY by combining the results
- \(\mathrm{P(x) = (3x^2 - x - 10) + (2x^2 - 9x + 4)}\)
- Combine like terms: \(\mathrm{3x^2 + 2x^2 = 5x^2}\), \(\mathrm{-x + (-9x) = -10x}\), \(\mathrm{-10 + 4 = -6}\)
- \(\mathrm{P(x) = 5x^2 - 10x - 6}\)
6. INFER the final answer
- From \(\mathrm{P(x) = 5x^2 - 10x - 6}\), we have \(\mathrm{a = 5}\), \(\mathrm{b = -10}\), \(\mathrm{c = -6}\)
- Therefore: \(\mathrm{a + b + c = 5 + (-10) + (-6) = -11}\)
Answer: -11
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Sign errors during binomial multiplication, especially with the negative terms.
Students often make mistakes like:
- \(\mathrm{(x - 2)(3x + 5) = 3x^2 + 5x - 6x + 10}\) (wrong sign on the last term)
- \(\mathrm{(2x - 1)(x - 4) = 2x^2 - 8x + x - 4}\) (wrong sign on the middle term)
These arithmetic errors cascade through the problem, leading to wrong coefficients and ultimately an incorrect sum.
Second Most Common Error:
Missing the shortcut INFER: Not recognizing that \(\mathrm{P(1) = a + b + c}\).
Even if students expand correctly, they might not realize this elegant verification method exists. While this doesn't cause a wrong answer, it shows incomplete mathematical understanding and misses a chance to double-check their work efficiently.
The Bottom Line:
This problem tests careful algebraic manipulation more than deep conceptual understanding. Success requires methodical expansion and meticulous attention to signs during arithmetic operations.