The area of a rectangular garden is represented by the polynomial 8x^2 + 3x + 10. Inside the garden, a...
GMAT Advanced Math : (Adv_Math) Questions
The area of a rectangular garden is represented by the polynomial \(8\mathrm{x}^2 + 3\mathrm{x} + 10\). Inside the garden, a square fountain has an area represented by the polynomial \(\mathrm{x}^2 - 2\mathrm{x} + 4\). If the remaining area is a patio, which of the following expressions represents the area of the patio?
1. TRANSLATE the problem setup
- Given information:
- Garden area: \(8x^2 + 3x + 10\)
- Fountain area: \(x^2 - 2x + 4\)
- Need: Patio area
- What this tells us: Patio area = Garden area - Fountain area
2. TRANSLATE into mathematical expression
- Set up the subtraction:
\(\text{Area of Patio} = (8x^2 + 3x + 10) - (x^2 - 2x + 4)\)
3. SIMPLIFY by distributing the negative sign
- When subtracting a polynomial, distribute the negative to every term in the second polynomial:
\(= 8x^2 + 3x + 10 - x^2 + 2x - 4\) - Notice how -2x becomes +2x and +4 becomes -4
4. SIMPLIFY by combining like terms
- Group terms with the same variable and power:
\(= (8x^2 - x^2) + (3x + 2x) + (10 - 4)\) - Combine the coefficients:
\(= 7x^2 + 5x + 6\)
Answer: C (\(7x^2 + 5x + 6\))
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY skill: Incorrectly distributing the negative sign when subtracting the second polynomial.
Students often write: \((8x^2 + 3x + 10) - (x^2 - 2x + 4) = 8x^2 + 3x + 10 - x^2 - 2x + 4\)
They forget that subtracting a negative makes it positive, so -2x should become +2x. This error gives them \(7x^2 + x + 14\), leading them to select Choice B (\(7x^2 + x + 14\)).
Second Most Common Error:
Weak SIMPLIFY skill: Making arithmetic mistakes when combining like terms, particularly with the constant terms.
Students correctly distribute the negative sign but then incorrectly combine 10 - 4 = 14 instead of 6, or make similar errors with the x terms. This could lead them to select Choice D (\(7x^2 + 5x + 14\)) or other incorrect options.
The Bottom Line:
This problem tests careful execution of polynomial subtraction - the setup is straightforward, but students must systematically distribute the negative sign and accurately combine like terms without rushing through the algebra.