Question:\(\mathrm{p(x) = x^3 + bx^2 + cx - 90}\)The polynomial function p is defined above, where b and c are...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{p(x) = x^3 + bx^2 + cx - 90}\)
The polynomial function p is defined above, where b and c are constants. If two of the zeros of the function are \(\mathrm{-6}\) and \(\mathrm{3}\), what is the value of the third zero?
- -15
- -5
- 5
- 10
- 15
1. TRANSLATE the problem information
- Given information:
- Polynomial: \(\mathrm{p(x) = x^3 + bx^2 + cx - 90}\)
- Two known zeros: -6 and 3
- Need to find the third zero
2. INFER the best approach
- Since we have a cubic polynomial and know two of its three zeros, Vieta's formulas provide the most direct path
- For a cubic \(\mathrm{p(x) = ax^3 + bx^2 + cx + d}\), the product of all roots equals \(\mathrm{-d/a}\)
- This means: (first zero) × (second zero) × (third zero) = \(\mathrm{-(-90)/1 = 90}\)
3. SIMPLIFY to find the third zero
- Set up the equation: \(\mathrm{(-6) \times (3) \times (third\ zero) = 90}\)
- Calculate the known product: \(\mathrm{(-6) \times (3) = -18}\)
- Solve: \(\mathrm{-18 \times (third\ zero) = 90}\)
- Therefore: third zero = \(\mathrm{90/(-18) = -5}\)
Answer: B) -5
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize that Vieta's formulas apply here and instead try to reconstruct the entire polynomial by finding coefficients b and c first.
This leads to a much more complex approach involving systems of equations, often causing students to make calculation errors or run out of time. They may abandon the systematic solution and guess, or select an incorrect answer due to computational mistakes.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify the need to use Vieta's formula but make sign errors.
The most common mistake is computing \(\mathrm{(-6) \times (3) \times r_3 = -90}\) instead of 90, forgetting that the product of roots equals \(\mathrm{-d/a = -(-90)/1 = +90}\). This leads them to solve \(\mathrm{-18r_3 = -90}\), getting \(\mathrm{r_3 = 5}\), causing them to select Choice C (5).
The Bottom Line:
This problem tests whether students can efficiently connect polynomial properties to direct solution methods rather than getting bogged down in unnecessary algebraic manipulation.