Which expression is equivalent to 12x^3 - 5x^3?

1. INFER what type of problem this is

• Given expression: \(12\mathrm{x}^3 - 5\mathrm{x}^3\)
• Both terms have the same variable part: \(\mathrm{x}^3\)
• This means they are like terms that can be combined

2. SIMPLIFY by combining the like terms

• When combining like terms, operate on the coefficients only
• Keep the variable part (\(\mathrm{x}^3\)) unchanged
• Perform the subtraction: \(12\mathrm{x}^3 - 5\mathrm{x}^3 = (12 - 5)\mathrm{x}^3 = 7\mathrm{x}^3\)

Answer: C. \(7\mathrm{x}^3\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing these as like terms, or thinking that combining terms somehow affects the exponents.

Some students incorrectly believe that when you combine terms with the same variable, you add the exponents (like in multiplication rules). They might think \(12\mathrm{x}^3 - 5\mathrm{x}^3\) becomes \(7\mathrm{x}^6\).

This may lead them to select Choice A (\(7\mathrm{x}^6\)).

Second Most Common Error:

Poor SIMPLIFY execution: Performing addition instead of subtraction on the coefficients.

Students might see the two terms and automatically add the coefficients: \(12 + 5 = 17\), giving \(17\mathrm{x}^3\). This often happens when students work quickly and don't carefully track the subtraction sign.

This may lead them to select Choice B (\(17\mathrm{x}^3\)).

The Bottom Line:

Success on this problem requires clearly distinguishing between combining like terms (where you operate on coefficients) and multiplying variables (where you add exponents). The key insight is that the variable part stays completely unchanged when combining like terms.