\((3\mathrm{y}^3 + 2\mathrm{y})(\mathrm{y}^3 - 2\mathrm{y})\)Which of the following is equivalent to the expression above?
GMAT Advanced Math : (Adv_Math) Questions
\((3\mathrm{y}^3 + 2\mathrm{y})(\mathrm{y}^3 - 2\mathrm{y})\)
Which of the following is equivalent to the expression above?
\(3\mathrm{y}^6 - 2\mathrm{y}^4 - 4\mathrm{y}^2\)
\(3\mathrm{y}^6 - 4\mathrm{y}^4 - 4\mathrm{y}^2\)
\(5\mathrm{y}^6 - 4\mathrm{y}^4 - 4\mathrm{y}^2\)
\(3\mathrm{y}^6 + 4\mathrm{y}^4 - 4\mathrm{y}^2\)
\(3\mathrm{y}^6 - 6\mathrm{y}^4 + 2\mathrm{y}^4 - 4\mathrm{y}^2\)
1. INFER the approach needed
- We have two binomials to multiply: \((3\mathrm{y}^3 + 2\mathrm{y})(\mathrm{y}^3 - 2\mathrm{y})\)
- Strategy: Use the distributive property to multiply each term in the first binomial by each term in the second binomial
2. SIMPLIFY using the distributive property
- Multiply each term systematically:
- \(3\mathrm{y}^3 \times \mathrm{y}^3 = 3\mathrm{y}^6\)
- \(3\mathrm{y}^3 \times (-2\mathrm{y}) = -6\mathrm{y}^4\)
- \(2\mathrm{y} \times \mathrm{y}^3 = 2\mathrm{y}^4\)
- \(2\mathrm{y} \times (-2\mathrm{y}) = -4\mathrm{y}^2\)
- This gives us: \(3\mathrm{y}^6 - 6\mathrm{y}^4 + 2\mathrm{y}^4 - 4\mathrm{y}^2\)
3. SIMPLIFY by combining like terms
- Identify like terms: \(-6\mathrm{y}^4\) and \(+2\mathrm{y}^4\) both have \(\mathrm{y}^4\)
- Combine: \(-6\mathrm{y}^4 + 2\mathrm{y}^4 = -4\mathrm{y}^4\)
- Final result: \(3\mathrm{y}^6 - 4\mathrm{y}^4 - 4\mathrm{y}^2\)
Answer: B (\(3\mathrm{y}^6 - 4\mathrm{y}^4 - 4\mathrm{y}^2\))
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students correctly expand the product but make arithmetic errors when combining like terms.
For example, they might calculate \(-6\mathrm{y}^4 + 2\mathrm{y}^4 = -2\mathrm{y}^4\) instead of \(-4\mathrm{y}^4\). This leads them to select Choice A (\(3\mathrm{y}^6 - 2\mathrm{y}^4 - 4\mathrm{y}^2\)).
Second Most Common Error:
Incomplete SIMPLIFY process: Students expand the binomials correctly but forget to combine like terms completely.
They stop at \(3\mathrm{y}^6 - 6\mathrm{y}^4 + 2\mathrm{y}^4 - 4\mathrm{y}^2\) without combining the middle terms. This leads them to select Choice E (\(3\mathrm{y}^6 - 6\mathrm{y}^4 + 2\mathrm{y}^4 - 4\mathrm{y}^2\)).
The Bottom Line:
This problem tests careful algebraic manipulation. Success requires both systematic expansion and precise arithmetic when combining like terms - two separate skills that both must be executed correctly.
\(3\mathrm{y}^6 - 2\mathrm{y}^4 - 4\mathrm{y}^2\)
\(3\mathrm{y}^6 - 4\mathrm{y}^4 - 4\mathrm{y}^2\)
\(5\mathrm{y}^6 - 4\mathrm{y}^4 - 4\mathrm{y}^2\)
\(3\mathrm{y}^6 + 4\mathrm{y}^4 - 4\mathrm{y}^2\)
\(3\mathrm{y}^6 - 6\mathrm{y}^4 + 2\mathrm{y}^4 - 4\mathrm{y}^2\)