The equation (x^2 + 6x - 7)/(x + 7) = ax + d is true for all x neq -7,...

1. TRANSLATE the problem information

• Given: \(\frac{x^2 + 6x - 7}{x + 7} = ax + d\) for all \(x \neq -7\)
• Find: \(a + d\) where \(a\) and \(d\) are integers

2. SIMPLIFY the rational expression by factoring

• Factor the numerator \(x^2 + 6x - 7\)
• Need two numbers that multiply to -7 and add to 6
• Those numbers are 7 and -1: \((7)(-1) = -7\) and \(7 + (-1) = 6\)
• So \(x^2 + 6x - 7 = (x + 7)(x - 1)\)

3. SIMPLIFY by canceling common factors

• Rewrite the equation: \(\frac{(x + 7)(x - 1)}{x + 7} = ax + d\)
• Since \(x \neq -7\), we know \(x + 7 \neq 0\), so we can cancel the common factor
• This gives us: \(x - 1 = ax + d\)

4. INFER the coefficient relationships

• For this equation to be true for all \(x \neq -7\), the polynomials on both sides must be identical
• Compare \(x - 1\) with \(ax + d\):
  • Coefficient of \(x\): \(a = 1\)
  • Constant term: \(d = -1\)

5. Calculate the final answer

• \(a + d = 1 + (-1) = 0\)

Answer: C. 0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students struggle with factoring the quadratic \(x^2 + 6x - 7\), either getting the wrong factors or not recognizing the factorization pattern at all.

Without proper factoring, they can't proceed with the cancellation step. Some might attempt polynomial long division instead, making the problem unnecessarily complex and error-prone. Others might guess at factoring and get something like \((x + 1)(x + 6)\) or other incorrect combinations.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor INFER reasoning: Students successfully factor and cancel to get \(x - 1 = ax + d\), but then don't recognize that they need to match coefficients systematically.

Instead of comparing coefficient by coefficient, they might substitute specific values of \(x\) (like \(x = 0\) or \(x = 1\)) to try to solve for \(a\) and \(d\). While this can work, it's more prone to arithmetic errors and doesn't reinforce the conceptual understanding of polynomial identity.

This may lead them to select Choice A (-6) or Choice B (-1) depending on their calculation errors.

The Bottom Line:

This problem requires strong algebraic manipulation skills combined with understanding that polynomial identities require coefficient-by-coefficient equality. The factoring step is the critical gateway - without it, students can't access the elegant cancellation that makes the problem solvable.