Unit 3 · Topic 02 · Algebra
Meera cut a rectangle of graph paper into four smaller rectangles with one horizontal and one vertical cut.
"Add up the four areas," she told Arjun, "and you've just multiplied two polynomials without writing a single times sign."
Arjun counted the squares. Then he tried it with algebra instead of numbers, and the same four pieces appeared.
After the identities practice, Meera and Arjun moved on to a harder-looking problem: multiply (x + 4) by (2x + 3). Arjun stared at it. "Two brackets, four terms total. Where do I even start?"
Meera took a sheet of graph paper and drew a big rectangle. She labelled one side (x + 4) and split it into two pieces of length x and 4. She labelled the other side (2x + 3) and split that into 2x and 3. The rectangle was now four smaller rectangles.
"Each small rectangle is a term," she said. "Top-left is x times 2x, which is 2x². Top-right is x times 3, which is 3x. Bottom-left is 4 times 2x, which is 8x. Bottom-right is 4 times 3, which is 12."
Arjun added the four pieces: 2x² + 3x + 8x + 12. He combined the like terms, 3x and 8x, to get 2x² + 11x + 12. "So multiplying two brackets is just multiplying every term in the first by every term in the second, and then tidying up."
"That's called distributing," Meera said. "People shorten it to FOIL for two-term brackets — First, Outer, Inner, Last — but that's just a memory trick for exactly what the rectangle already shows you. It stops working the moment a bracket has three terms, which is why the rectangle method is worth understanding properly."
To prove it, she gave him (x + 2)(x² + 3x + 1) — three terms in the second bracket. Arjun drew a rectangle with three columns instead of two, distributed x and 2 across all three terms, and collected six pieces into x³ + 3x² + x + 2x² + 6x + 2, then combined the x² terms to land on x³ + 5x² + 7x + 2.
"FOIL would've had no letter for that," he said. "But the rectangle just needed one more column."
Meera nodded. "Multiplying polynomials is never really a new rule. It's always the same one — every term touches every term — however many terms you've got."
To multiply two polynomials, multiply every term of the first by every term of the second, then combine like terms. This is called the distributive law, and it works no matter how many terms each polynomial has.
For two two-term brackets, this is sometimes remembered as FOIL — First, Outer, Inner, Last — but FOIL is just a special case of distributing that breaks down once a bracket has three or more terms.
A rectangle-area picture makes the rule visible: split a rectangle's two sides into the terms of each polynomial, and the areas of the resulting smaller rectangles are exactly the products you need to add up. This also explains WHY it works — area doesn't care how you split the rectangle, so the total must be the same either way.
Always finish by combining like terms — terms with the same letter part raised to the same power — since an un-simplified answer is not considered complete.
Draw a rectangle on graph paper, split one side into 3 units + 2 units and the other into 4 units + 1 unit. Find the total area two ways: as one big rectangle, and as four small ones added up — confirm they match.
Multiply (x + 1)(x + 1)(x + 1) by first multiplying any two brackets, then multiplying that answer by the third bracket.
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