Question
Multiply and simplify.
Solution — Step by Step
We need to multiply every term in by every term in .
The first bracket has 2 terms, the second has 3 terms. This gives partial products in total.
Group by degree:
- terms:
- terms:
- terms:
- Constant:
Why This Works
Polynomial multiplication is just the distributive property applied repeatedly. Each term in the first polynomial “visits” each term in the second. The number of visits equals (terms in first) × (terms in second).
The key skill is the final collecting step. Writing partial products vertically (aligned by degree) reduces errors:
2x³ - 4x² + 2x
3x² - 6x + 3
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2x³ - x² - 4x + 3For a neat layout in exams, write the smaller polynomial on the left and the larger one on the right. After all partial products are written, underline each group of like terms with the same colour before collecting.
Alternative Method
You can also use long polynomial multiplication, written column by column — similar to multi-digit number multiplication:
Both methods give the same result. Choose whichever keeps you organised.
Common Mistake
Students often multiply only the “adjacent” terms (outer × outer, inner × inner, like FOIL for two binomials) and miss the cross-products. FOIL only works for binomial × binomial (2 × 2 = 4 products). For binomial × trinomial (2 × 3 = 6 products), you must distribute every term in the first bracket across all three terms in the second. Skipping even one cross-product gives a wrong answer.