Which of the following is equivalent to the sum of 3x^4 + 2x^3 and 4x^4 + 7x^3?

1. TRANSLATE the problem information

• Given: We need to find the sum of \(\mathrm{3x^4 + 2x^3}\) and \(\mathrm{4x^4 + 7x^3}\)
• This means: \(\mathrm{3x^4 + 2x^3 + 4x^4 + 7x^3}\)

2. INFER the approach

• Strategy: Look for like terms that can be combined
• Like terms have the same variable raised to the same power
• I can see \(\mathrm{x^4}\) terms and \(\mathrm{x^3}\) terms that can be grouped together

3. SIMPLIFY by combining like terms

• Group \(\mathrm{x^4}\) terms: \(\mathrm{3x^4 + 4x^4 = 7x^4}\)
• Group \(\mathrm{x^3}\) terms: \(\mathrm{2x^3 + 7x^3 = 9x^3}\)
• Final result: \(\mathrm{7x^4 + 9x^3}\)

Answer: D. \(\mathrm{7x^4 + 9x^3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak conceptual understanding of exponent operations: Students incorrectly think they should add the exponents when combining like terms, treating this like multiplication rules instead of addition rules.

For example, they might think \(\mathrm{3x^4 + 4x^4}\) becomes \(\mathrm{7x^8}\) (adding both coefficients AND exponents). This leads them to calculate \(\mathrm{7x^8 + 9x^6}\) instead of the correct \(\mathrm{7x^4 + 9x^3}\).

This may lead them to select Choice B (\(\mathrm{7x^8 + 9x^6}\))

Second Most Common Error:

Poor SIMPLIFY execution with coefficient operations: Students multiply coefficients instead of adding them when combining like terms.

They might calculate: \(\mathrm{3 \times 4 = 12}\) for the \(\mathrm{x^4}\) terms and \(\mathrm{2 \times 7 = 14}\) for the \(\mathrm{x^3}\) terms, getting \(\mathrm{12x^4 + 14x^3}\).

This may lead them to select Choice C (\(\mathrm{12x^4 + 14x^3}\))

The Bottom Line:

This problem tests whether students understand that polynomial addition only affects coefficients, not exponents, and that "combining like terms" means adding coefficients, not multiplying them.