Which expression is equivalent to \(3\mathrm{x}(4\mathrm{x} + 7)\)?

Part 1: Brief Solution

Concepts Tested: Factoring algebraic expressions, finding greatest common factor

Primary Process Skills: Simplify, Translate

Essential Steps:

• Identify that both terms \(\mathrm{9x^2}\) and \(\mathrm{5x}\) contain the common factor \(\mathrm{x}\)
• Factor out \(\mathrm{x}\) to get \(\mathrm{x(9x + 5)}\)
• Verify by expanding back to the original expression
• Match with answer choice A

Answer: A

Part 2: Top 3 Faltering Points

Top 3 Faltering Points:

1. Partial Factoring Error - Phase: Executing Approach → Choice B (\(\mathrm{5x(9x + 1)}\))
   • Process skill failure: Simplify
   • Student factors out \(\mathrm{5x}\) instead of recognizing \(\mathrm{x}\) as the greatest common factor.
2. Distribution Confusion - Phase: Devising Approach → Choice C (\(\mathrm{9x(x + 5)}\))
   • Process skill failure: Translate
   • Student incorrectly assumes \(\mathrm{9x}\) can be factored out, creating \(\mathrm{9x^2 + 45x}\) instead of \(\mathrm{9x^2 + 5x}\).
3. Degree Misunderstanding - Phase: Executing Approach → Choice D (\(\mathrm{x^2(9x + 5)}\))
   • Process skill failure: Simplify
   • Student mistakenly factors out \(\mathrm{x^2}\) instead of \(\mathrm{x}\), creating a cubic expression instead of quadratic.

Part 3: Detailed Solution

Understanding the Question:

We need to find an equivalent expression to \(\mathrm{9x^2 + 5x}\) by factoring.

Process Skill: TRANSLATE - "Equivalent" means the expressions have the same value for all values of \(\mathrm{x}\), and factoring is the reverse process of distributing.

Step 1: Identify the Greatest Common Factor

Look at each term:

• \(\mathrm{9x^2 = 9 \times x \times x}\)
• \(\mathrm{5x = 5 \times x}\)

Process Skill: SIMPLIFY - We need to find what factor both terms share. Both terms contain the variable \(\mathrm{x}\) as a factor, but \(\mathrm{9x^2}\) has \(\mathrm{x^2}\) while \(\mathrm{5x}\) has just \(\mathrm{x}\). The greatest common factor is \(\mathrm{x}\) (the highest power of \(\mathrm{x}\) that divides both terms).

Step 2: Factor Out the GCF

When we factor out \(\mathrm{x}\) from each term:

\(\mathrm{9x^2 + 5x = x(9x) + x(5)}\)

Process Skill: SIMPLIFY - Using the distributive property in reverse, we can write this as:

\(\mathrm{= x(9x + 5)}\)

Step 3: Verification

Let's verify our answer by expanding \(\mathrm{x(9x + 5)}\):

\(\mathrm{x(9x + 5)}\)

\(\mathrm{= x \times 9x + x \times 5}\)

\(\mathrm{= 9x^2 + 5x}\) ✓

This matches our original expression, confirming our answer.

Process Skill: INFER - Since our factored form expands back to the original expression, we can conclude they are equivalent.

Checking Other Choices:

• Choice B: \(\mathrm{5x(9x + 1) = 45x^2 + 5x \neq 9x^2 + 5x}\)
• Choice C: \(\mathrm{9x(x + 5) = 9x^2 + 45x \neq 9x^2 + 5x}\)
• Choice D: \(\mathrm{x^2(9x + 5) = 9x^3 + 5x^2 \neq 9x^2 + 5x}\)

Answer: A

Part 4: Detailed Faltering Points Analysis

Errors while devising the approach:

• Misunderstanding Factoring Direction: Some students might try to expand rather than factor, not recognizing that we need to find a factored form equivalent to the given expression.
• GCF Misconception: Students might not understand how to identify the greatest common factor when dealing with variables and coefficients together.

Errors while executing the approach:

• Partial Factoring (Choice B): Students factor out \(\mathrm{5x}\) instead of just \(\mathrm{x}\), thinking that since \(\mathrm{5x}\) appears as a complete term, it must be the GCF. This leads to \(\mathrm{5x(9x + 1) = 45x^2 + 5x}\), which doesn't match the original.
• Over-factoring (Choice D): Students factor out \(\mathrm{x^2}\) thinking "more factoring is better," resulting in \(\mathrm{x^2(9x + 5) = 9x^3 + 5x^2}\), which changes the degree of the polynomial.
• Coefficient Confusion (Choice C): Students incorrectly assume they can factor out \(\mathrm{9x}\), leading to \(\mathrm{9x(x + 5) = 9x^2 + 45x}\), where the second term becomes \(\mathrm{45x}\) instead of \(\mathrm{5x}\).

Errors while selecting the answer:

• Verification Skipping: Students might not verify their factored form by expanding it back, missing computational errors.
• Pattern Recognition Error: Students might select an answer that "looks similar" without careful calculation, especially if they see familiar patterns like \(\mathrm{x}\)(something) or coefficient(\(\mathrm{x}\))(something).