Expand (x + 3)(x² − 2x + 1). Which result is correct?
Ax³ − 2x² + x + 3 (distributing only the x-term and then adding 3)
Bx³ + x² − 5x + 3
Cx³ + x² + x + 3
Dx³ − 2x² + 3x² + 1
Distributing fully: x·x² = x³, x·(−2x) = −2x², x·1 = x; then 3·x² = 3x², 3·(−2x) = −6x, 3·1 = 3. Combining like terms: x³ + (−2+3)x² + (1−6)x + 3 = x³ + x² − 5x + 3. Option A is the classic error — the 3 is only added once rather than distributed across all three terms of the trinomial.
Question 2 Multiple Choice
When you multiply a binomial by a trinomial using the 'each by each' approach, how many individual products are generated before combining like terms?
A3 — one for each term in the trinomial
B4 — like the FOIL method
C5 — the sum of the number of terms in both polynomials
D6 — the product of the number of terms in each polynomial
Each of the 2 terms in the binomial multiplies each of the 3 terms in the trinomial: 2 × 3 = 6 products total. This generalizes: an m-term polynomial times an n-term polynomial always produces m × n products before combining. FOIL's four products are simply the 2 × 2 case.
Question 3 True / False
When computing 3x · 2x², the exponents 1 and 2 are added to give x³.
TTrue
FFalse
Answer: True
The product rule for exponents states x^a · x^b = x^(a+b). Here, x¹ · x² = x^(1+2) = x³, and the coefficients 3 and 2 multiply separately to give 6, yielding 6x³. This exponent rule applies at every individual multiplication step in polynomial multiplication.
Question 4 True / False
FOIL is a reliable method for multiplying any two polynomials.
TTrue
FFalse
Answer: False
FOIL (First, Outer, Inner, Last) only works for binomial × binomial — the 2 × 2 case. It breaks down the moment either polynomial has more than two terms. For a binomial times a trinomial (2 × 3), you need 6 products, but FOIL only produces 4. The general 'each by each' approach or grid model works for any polynomial sizes.
Question 5 Short Answer
Why does the grid (area) model prevent the most common error in polynomial multiplication?
Think about your answer, then reveal below.
Model answer: The grid assigns exactly one cell to each pair of terms — one row per term in the first polynomial, one column per term in the second. Every cell must be filled, making it structurally impossible to skip a term. You cannot accidentally 'forget' to multiply 3 by −2x because that cell exists in the grid and must be computed.
The most common error is partial distribution — distributing one term to all others, but missing some combinations for another term. The grid makes every combination explicit and visible, turning a procedural task (distribute everything) into a spatial one (fill every cell). This is also why the grid reveals the structural parallel to multi-digit integer multiplication.