Which of the following expressions is equivalent to 8x^(10) - 8x^9 + 88x?
GMAT Advanced Math : (Adv_Math) Questions
Which of the following expressions is equivalent to \(8\mathrm{x}^{10} - 8\mathrm{x}^9 + 88\mathrm{x}\)?
1. INFER the problem strategy
- Given expression: \(8\mathrm{x}^{10} - 8\mathrm{x}^9 + 88\mathrm{x}\)
- This is a factoring problem - I need to find the greatest common factor (GCF) of all terms
2. SIMPLIFY by finding the GCF of coefficients
- Coefficients: 8, -8, 88
- GCF of coefficients = 8
3. SIMPLIFY by finding the GCF of variables
- Variable parts: \(\mathrm{x}^{10}\), \(\mathrm{x}^9\), \(\mathrm{x}^1\)
- GCF of variables = x (the lowest power)
4. SIMPLIFY by factoring out the complete GCF
- Complete GCF = \(8\mathrm{x}\)
- Factor out \(8\mathrm{x}\): \(8\mathrm{x}^{10} - 8\mathrm{x}^9 + 88\mathrm{x} = 8\mathrm{x}(\mathrm{x}^9 - \mathrm{x}^8 + 11)\)
- Check each term:
- \(8\mathrm{x}^{10} ÷ 8\mathrm{x} = \mathrm{x}^9\) ✓
- \(8\mathrm{x}^9 ÷ 8\mathrm{x} = \mathrm{x}^8\) ✓
- \(88\mathrm{x} ÷ 8\mathrm{x} = 11\) ✓
Answer: D. \(8\mathrm{x}(\mathrm{x}^9 - \mathrm{x}^8 + 11)\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make exponent arithmetic errors when dividing terms by the GCF. For example, they might incorrectly calculate \(8\mathrm{x}^{10} ÷ 8\mathrm{x} = \mathrm{x}^{10}\) instead of \(\mathrm{x}^9\), or \(8\mathrm{x}^9 ÷ 8\mathrm{x} = \mathrm{x}^9\) instead of \(\mathrm{x}^8\).
This leads them to arrive at an incorrect factored form that doesn't match any of the given choices, causing confusion and guessing.
Second Most Common Error:
Incomplete INFER reasoning: Students might only factor out part of the GCF (like just 8 or just x) instead of recognizing that \(8\mathrm{x}\) is the complete greatest common factor.
This may lead them to select Choice C (\(8\mathrm{x}(\mathrm{x}^{10} - \mathrm{x}^9 + 11\mathrm{x})\)) because they factored out \(8\mathrm{x}\) but didn't properly reduce the exponents in each term.
The Bottom Line:
This problem tests whether students can systematically identify and factor out the complete greatest common factor while maintaining precision with exponent arithmetic.