A rectangular garden plot has dimensions that can be represented by the polynomial \(\mathrm{x^3(4x^2 - 5x + 6)}\). Which expression...
GMAT Advanced Math : (Adv_Math) Questions
- \(\mathrm{4x^5 - 5x^4 + 6}\)
- \(\mathrm{4x^5 - 5x^4 + 6x^3}\)
- \(\mathrm{4x^5 + 5x^4 + 6x^3}\)
- \(\mathrm{4x^6 - 5x^4 + 6x^3}\)
1. TRANSLATE the problem information
- Given information:
- Garden plot dimensions represented by \(\mathrm{x^3(4x^2 - 5x + 6)}\)
- Need to find the total area expression
- This tells us we need to expand/multiply out this polynomial expression.
2. SIMPLIFY by distributing x³ to each term
- Use the distributive property: \(\mathrm{x^3(4x^2 - 5x + 6)}\) means multiply \(\mathrm{x^3}\) by each term inside the parentheses
- First term: \(\mathrm{x^3 \cdot 4x^2 = 4x^{3+2} = 4x^5}\)
- Second term: \(\mathrm{x^3 \cdot (-5x) = -5x^{3+1} = -5x^4}\)
- Third term: \(\mathrm{x^3 \cdot 6 = 6x^3}\)
3. SIMPLIFY to write the final expanded form
- Combine all terms: \(\mathrm{4x^5 - 5x^4 + 6x^3}\)
Answer: B (\(\mathrm{4x^5 - 5x^4 + 6x^3}\))
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students forget to distribute \(\mathrm{x^3}\) to ALL terms in the parentheses, especially the constant term 6.
They might only multiply \(\mathrm{x^3}\) by the first two terms (\(\mathrm{4x^2}\) and \(\mathrm{-5x}\)), giving them \(\mathrm{4x^5 - 5x^4}\), and then just add 6 at the end instead of multiplying it by \(\mathrm{x^3}\).
This leads them to select Choice A (\(\mathrm{4x^5 - 5x^4 + 6}\)).
Second Most Common Error:
Poor SIMPLIFY reasoning: Students make sign errors or exponent mistakes during distribution.
They might incorrectly handle the negative sign in \(\mathrm{-5x}\), changing it to positive, or make errors in adding exponents (like thinking \(\mathrm{x^3 \cdot x^2 = x^6}\) instead of \(\mathrm{x^5}\)).
This confusion can lead them to select Choice C (\(\mathrm{4x^5 + 5x^4 + 6x^3}\)) or Choice D (\(\mathrm{4x^6 - 5x^4 + 6x^3}\)).
The Bottom Line:
This problem tests careful application of the distributive property combined with exponent rules. Students must systematically distribute to every term and correctly apply \(\mathrm{x^a \cdot x^b = x^{a+b}}\) while maintaining proper signs throughout.