\(\mathrm{f(x) = x^2 + bx}\)\(\mathrm{g(x) = 9x^2 - 27x}\)Functions f and g are given, and in function f, b is...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = x^2 + bx}\) (where b is unknown)
  • \(\mathrm{g(x) = 9x^2 - 27x}\)
  • \(\mathrm{f(x) \cdot g(x) = 9x^4 - 26x^3 - 3x^2}\)
• What we need to find: The value of constant b

2. INFER the solution strategy

• Since we know what \(\mathrm{f(x) \cdot g(x)}\) equals, we can multiply the given functions and compare the result
• By comparing coefficients of like terms, we can set up equations to solve for b
• This approach will give us the exact value we need

3. SIMPLIFY the polynomial multiplication

• Multiply \(\mathrm{f(x)}\) and \(\mathrm{g(x)}\):
  \(\mathrm{f(x) \cdot g(x) = (x^2 + bx)(9x^2 - 27x)}\)
• Apply distributive property:
  \(\mathrm{= x^2(9x^2 - 27x) + bx(9x^2 - 27x)}\)
  \(\mathrm{= 9x^4 - 27x^3 + 9bx^3 - 27bx^2}\)
• Combine like terms:
  \(\mathrm{= 9x^4 + (-27 + 9b)x^3 + (-27b)x^2}\)

4. INFER coefficient relationships

• Set our result equal to the given expression:
  \(\mathrm{9x^4 + (-27 + 9b)x^3 + (-27b)x^2 = 9x^4 - 26x^3 - 3x^2}\)
• For polynomials to be equal, corresponding coefficients must be equal:
  • \(\mathrm{x^4}\) coefficient: \(\mathrm{9 = 9}\) ✓
  • \(\mathrm{x^3}\) coefficient: \(\mathrm{-27 + 9b = -26}\)
  • \(\mathrm{x^2}\) coefficient: \(\mathrm{-27b = -3}\)

5. SIMPLIFY to solve for b

• From the \(\mathrm{x^3}\) coefficient equation: \(\mathrm{-27 + 9b = -26}\)
  Add 27 to both sides: \(\mathrm{9b = 1}\)
  Divide by 9: \(\mathrm{b = 1/9}\)
• Let's verify with the \(\mathrm{x^2}\) coefficient: \(\mathrm{-27b = -3}\)
  Divide by -27: \(\mathrm{b = 1/9}\) ✓

Answer: C. 1/9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors during polynomial multiplication, especially when distributing negative terms like \(\mathrm{-27x}\).

A common mistake is writing:
\(\mathrm{(x^2 + bx)(9x^2 - 27x) = 9x^4 - 27x^3 + 9bx^3 + 27bx^2}\)

Notice the incorrect positive sign on the last term. This happens when students lose track of \(\mathrm{(-27x) \times (bx) = -27bx^2}\). With this error, the \(\mathrm{x^2}\) coefficient becomes \(\mathrm{+27b}\) instead of \(\mathrm{-27b}\), leading to the equation \(\mathrm{27b = -3}\), which gives \(\mathrm{b = -1/9}\). Since this exact value isn't among the choices, this leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students correctly multiply the polynomials but fail to systematically compare coefficients. Instead, they might try to solve by plugging in answer choices or attempt to factor the resulting expressions.

This approach becomes much more complicated and time-consuming. Students often abandon the systematic method partway through and resort to guessing, frequently gravitating toward Choice D (9) because it appears in the original functions.

The Bottom Line:

This problem rewards systematic polynomial manipulation and coefficient comparison. Students who rush through the distributive property steps or don't organize their coefficient comparison clearly are most likely to make errors that derail their solution.