Which of the following expressions is equivalent to the sum of \(\mathrm{(r^3 + 5r^2 + 7)}\) and \(\mathrm{(r^2 + 8r...

1. TRANSLATE the problem information

• We need to find the sum of two polynomial expressions:
  • First expression: \(\mathrm{(r^3 + 5r^2 + 7)}\)
  • Second expression: \(\mathrm{(r^2 + 8r + 12)}\)
• 'Sum' means we add these expressions together

2. TRANSLATE into mathematical notation

Set up the addition:

\(\mathrm{(r^3 + 5r^2 + 7) + (r^2 + 8r + 12)}\)

3. SIMPLIFY by removing parentheses

Since we're adding, we can remove the parentheses:

\(\mathrm{r^3 + 5r^2 + 7 + r^2 + 8r + 12}\)

4. INFER which terms can be combined

• Like terms have the same variable raised to the same power
• Group the like terms together:
  • \(\mathrm{r^3}\) terms: \(\mathrm{r^3}\) (only one)
  • \(\mathrm{r^2}\) terms: \(\mathrm{5r^2}\) and \(\mathrm{r^2}\)
  • \(\mathrm{r}\) terms: \(\mathrm{8r}\) (only one)
  • constant terms: 7 and 12

5. SIMPLIFY by combining like terms

• \(\mathrm{r^3}\) stays as \(\mathrm{r^3}\)
• \(\mathrm{r^2}\) terms: \(\mathrm{5r^2 + r^2 = 6r^2}\)
• \(\mathrm{r}\) terms: \(\mathrm{8r}\) stays as \(\mathrm{8r}\)
• Constants: \(\mathrm{7 + 12 = 19}\)

Answer: D. \(\mathrm{r^3 + 6r^2 + 8r + 19}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students incorrectly identify unlike terms as being combinable, such as trying to add \(\mathrm{r^3}\) and \(\mathrm{r^2}\) terms together, or adding \(\mathrm{r^2}\) and \(\mathrm{r}\) terms together.

For example, they might think \(\mathrm{5r^2 + 8r = 13r^2}\) (incorrectly treating \(\mathrm{r^2}\) and \(\mathrm{r}\) as the same type of term), leading them to get something like \(\mathrm{r^3 + 13r^2 + 19}\). This may lead them to select Choice B \(\mathrm{(2r^3 + 13r^2 + 19)}\) if they also make additional errors with the \(\mathrm{r^3}\) term.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the problem correctly but make arithmetic errors when combining the constant terms or coefficients, or fail to combine all like terms systematically.

They might correctly identify that \(\mathrm{5r^2 + r^2 = 6r^2}\), but then forget to include the \(\mathrm{8r}\) term or make an error with the constants, leading to incomplete expressions. This causes them to get confused and potentially select Choice C \(\mathrm{(r^3 + 5r^2 + 7r + 12)}\) which looks like they started correctly but didn't finish combining terms.

The Bottom Line:

This problem tests whether students truly understand what makes terms 'like terms' and can systematically organize and combine them. The key insight is that only terms with identical variable parts (same variables raised to same powers) can be added together.