Question:The equation x^3 - 3x^2 - 4x = 0 has three solutions. What is the product of these solutions?-4-1034

1. INFER the factoring strategy

• Given: \(\mathrm{x^3 - 3x^2 - 4x = 0}\)
• Key insight: Notice that every term contains x, so we can factor out x as a common factor
• This will give us one solution immediately (x = 0) and a quadratic to solve for the other two

2. SIMPLIFY by factoring out the common factor

• Factor out x: \(\mathrm{x(x^2 - 3x - 4) = 0}\)
• Now we have the product of x and a quadratic expression equal to zero

3. SIMPLIFY by factoring the quadratic expression

• We need to factor \(\mathrm{x^2 - 3x - 4}\)
• Look for two numbers that multiply to -4 and add to -3
• Testing factor pairs: (-4)(+1) = -4 and (-4) + (+1) = -3 ✓
• So \(\mathrm{x^2 - 3x - 4 = (x - 4)(x + 1)}\)

4. INFER solutions using zero product property

• Complete factorization: \(\mathrm{x(x - 4)(x + 1) = 0}\)
• From zero product property, each factor can equal zero:
  • \(\mathrm{x = 0}\)
  • \(\mathrm{x - 4 = 0 \Rightarrow x = 4}\)
  • \(\mathrm{x + 1 = 0 \Rightarrow x = -1}\)

5. SIMPLIFY to find the product

• Product of solutions: \(\mathrm{0 \times 4 \times (-1) = 0}\)

Answer: C (0)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly factor the quadratic \(\mathrm{x^2 - 3x - 4}\)

Many students struggle to find the correct factor pair. They might try combinations like (-2)(+2) or (-1)(+4), which don't work. Some might even get confused about the signs and try \(\mathrm{(x + 4)(x + 1)}\) or \(\mathrm{(x - 4)(x - 1)}\). When they expand these incorrect factorizations, they don't match the original quadratic, but they proceed anyway.

This leads them to find incorrect solutions and calculate the wrong product, potentially selecting Choice A (-4) or Choice E (4).

Second Most Common Error:

Weak INFER reasoning: Students forget that x = 0 is a solution

Some students see the equation and immediately try to divide everything by x, essentially "canceling out" the x factor. This eliminates the x = 0 solution entirely. They solve only the quadratic \(\mathrm{x^2 - 3x - 4 = 0}\) and find x = 4 and x = -1, then calculate the product as \(\mathrm{4 \times (-1) = -4}\).

This may lead them to select Choice A (-4).

The Bottom Line:

This problem tests your systematic factoring skills and attention to detail. The key insight is recognizing that factoring out the common factor x immediately gives you one solution, and you must remember to include it in your final calculation.