Which expression is equivalent to 5x^5 - 6x^4 + 8x^3?

1. INFER the problem type and strategy

• This is asking for an equivalent expression, which signals a factoring problem
• Strategy: Find the greatest common factor (GCF) of all terms and factor it out

2. INFER the greatest common factor

• Look at coefficients: 5, -6, 8
  • These have no common factors other than 1
• Look at variables: \(\mathrm{x^5}\), \(\mathrm{x^4}\), \(\mathrm{x^3}\)
  • The lowest power is \(\mathrm{x^3}\), so \(\mathrm{x^3}\) is the common factor
• Overall GCF = \(\mathrm{x^3}\)

3. SIMPLIFY by factoring out \(\mathrm{x^3}\)

• Divide each term by \(\mathrm{x^3}\):

• \(\mathrm{5x^5 ÷ x^3 = 5x^2}\)
• \(\mathrm{-6x^4 ÷ x^3 = -6x}\)
• \(\mathrm{8x^3 ÷ x^3 = 8}\)

• Result: \(\mathrm{x^3(5x^2 - 6x + 8)}\)

4. SIMPLIFY by checking your work

• Expand \(\mathrm{x^3(5x^2 - 6x + 8)}\) to verify:
• \(\mathrm{x^3(5x^2) - x^3(6x) + x^3(8) = 5x^5 - 6x^4 + 8x^3}\) ✓

Answer: B. \(\mathrm{x^3(5x^2 - 6x + 8)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that finding an "equivalent expression" means factoring, or they incorrectly identify the GCF by only looking at coefficients or only looking at variables.

For example, they might think the GCF is just 1 (ignoring the \(\mathrm{x^3}\)) or incorrectly choose \(\mathrm{x^4}\) or \(\mathrm{x^5}\) as the variable factor. This leads to incorrect factoring and may cause them to select Choice A (\(\mathrm{x^4(5x - 6)}\)) if they use \(\mathrm{x^4}\) as their factor, or leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify \(\mathrm{x^3}\) as the GCF but make arithmetic errors when dividing the coefficients or subtracting exponents.

They might incorrectly calculate \(\mathrm{5x^5 ÷ x^3 = 5x^3}\) instead of \(\mathrm{5x^2}\), or make similar mistakes with other terms. This creates an incorrect factored form that doesn't match any answer choice, causing them to get stuck and guess.

The Bottom Line:

This problem tests whether students can systematically identify the greatest common factor and execute the factoring process accurately. Success requires both strategic thinking (recognizing the factoring approach) and careful algebraic manipulation.