Which expression is equivalent to 15w^2 + 8w?
GMAT Advanced Math : (Adv_Math) Questions
Which expression is equivalent to \(15\mathrm{w}^2 + 8\mathrm{w}\)?
1. TRANSLATE the problem information
- Given: The expression \(15\mathrm{w}^2 + 8\mathrm{w}\)
- Find: Which answer choice is equivalent to this expression
2. INFER the solution approach
- Both terms in \(15\mathrm{w}^2 + 8\mathrm{w}\) contain the variable w
- Look for a common factor that can be factored out
- Factoring will create an equivalent expression
3. SIMPLIFY by factoring out the common factor
- Identify the common factor: Both \(15\mathrm{w}^2\) and \(8\mathrm{w}\) contain w
- Factor out w: \(15\mathrm{w}^2 + 8\mathrm{w} = \mathrm{w}(15\mathrm{w} + 8)\)
- This gives us \(\mathrm{w}(15\mathrm{w} + 8)\)
4. SIMPLIFY by checking answer choices
- Choice A: \(\mathrm{w}(15\mathrm{w} + 8)\) - This matches our factored form ✓
- Choice B: \(8\mathrm{w}(15\mathrm{w} + 1) = 120\mathrm{w}^2 + 8\mathrm{w} ≠ 15\mathrm{w}^2 + 8\mathrm{w}\)
- Choice C: \(15\mathrm{w}^2(2\mathrm{w} + 1) = 30\mathrm{w}^3 + 15\mathrm{w}^2 ≠ 15\mathrm{w}^2 + 8\mathrm{w}\)
- Choice D: \(23(\mathrm{w}^2 + \mathrm{w}) = 23\mathrm{w}^2 + 23\mathrm{w} ≠ 15\mathrm{w}^2 + 8\mathrm{w}\)
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY skill: Students may attempt to factor incorrectly, perhaps trying to factor each term separately instead of finding the common factor. For example, they might think \(15\mathrm{w}^2\) factors as \(15(\mathrm{w}^2)\) and \(8\mathrm{w}\) factors as \(8(\mathrm{w})\), missing that w can be factored from both terms together.
This confusion about proper factoring technique may lead them to select Choice D (\(23(\mathrm{w}^2 + \mathrm{w})\)) because they recognize both original terms appear but don't verify the coefficients are correct.
Second Most Common Error:
Poor INFER reasoning: Students may not recognize that they need to factor at all. Instead, they might try to manipulate the answer choices without a clear strategy, or attempt to combine like terms incorrectly (like trying to add \(15\mathrm{w}^2 + 8\mathrm{w} = 23\mathrm{w}^2\)).
This leads to confusion and guessing among the available choices.
The Bottom Line:
Success on this problem requires recognizing the factoring strategy and correctly identifying the greatest common factor. Students who struggle with factoring fundamentals or who don't verify their answer by expanding will have difficulty with this type of algebraic manipulation.