A student factoring x² + 8x + 15 reasons: 'I need two numbers that multiply to 8 and add to 15.' What is the error, and what is the correct factorization?
AThe student should use the AC method since the leading coefficient is not 1. The answer is (x + 3)(x + 5).
BThe student inverted the conditions. You need numbers that add to 8 and multiply to 15. The answer is (x + 3)(x + 5).
CThe student's approach is correct; since no integer pair works, the trinomial is irreducible over the integers.
DThe student should use the quadratic formula instead of attempting to factor by inspection.
In x² + bx + c, b comes from the sum of the two numbers and c comes from their product — because FOIL gives (x+p)(x+q) = x² + (p+q)x + pq. The most common error is reversing these roles. For x² + 8x + 15, we need p + q = 8 and p × q = 15. The pair (3, 5) satisfies both: 3 + 5 = 8 and 3 × 5 = 15, giving (x + 3)(x + 5).
Question 2 Multiple Choice
Which of the following is the correct factorization of x² − 3x − 10?
A(x + 5)(x + 2)
B(x − 5)(x − 2)
C(x − 5)(x + 2)
D(x + 5)(x − 2)
We need p + q = −3 and p × q = −10. Since c is negative, one factor must be positive and one negative. Testing: (−5) + 2 = −3 ✓ and (−5)(2) = −10 ✓. So (x − 5)(x + 2) is correct. Verify by FOILing: x² + 2x − 5x − 10 = x² − 3x − 10. Option D, (x + 5)(x − 2), gives x² + 3x − 10 — the sign of b is flipped, a common error when negatives are involved.
Question 3 True / False
When factoring x² − 9x + 20, you should look for two negative numbers that add to −9 and multiply to 20.
TTrue
FFalse
Answer: True
When c (the constant term) is positive and b (the middle coefficient) is negative, both numbers must be negative: a negative times a negative gives a positive product, and two negatives sum to a negative. For x² − 9x + 20, we need numbers that add to −9 and multiply to 20: that's −4 and −5, giving (x − 4)(x − 5). Forgetting to consider negative factors is one of the most common errors in factoring.
Question 4 True / False
When the leading coefficient is greater than 1 (e.g., 2x² + 7x + 3), you can factor the trinomial by finding two numbers that add to 7 and multiply to 3.
TTrue
FFalse
Answer: False
When a ≠ 1, you cannot simply find factors of c that add to b. The AC method is needed: multiply a × c (here, 2 × 3 = 6), then find factors of that product that add to b (here, 1 + 6 = 7). Use these to split the middle term: 2x² + x + 6x + 3, then factor by grouping: x(2x + 1) + 3(2x + 1) = (x + 3)(2x + 1). Applying the simpler method ignores the leading coefficient and produces incorrect results.
Question 5 Short Answer
Explain why factoring a trinomial of the form x² + bx + c is described as 'reversing FOIL,' and what two conditions the two numbers must satisfy.
Think about your answer, then reveal below.
Model answer: Factoring reverses FOIL because FOIL-ing (x + p)(x + q) produces x² + (p+q)x + pq — so b equals the sum p+q and c equals the product p×q. To factor, we find p and q such that p + q = b and p × q = c.
This is why b and c play different roles: b comes from adding the two numbers while c comes from multiplying them. A student who searches for numbers that 'add to c' or 'multiply to b' has inverted the relationship. Always verify by FOILing the answer: if the result matches the original trinomial, the factors are correct.