L2 · Binomial times binomial

Learning goal

“I can multiply two binomials using the area model or the distributive record, relate the product to multiplying two-digit numbers, and verify the product by substitution.”

Binomial — a polynomial with exactly two terms (e.g. x + 3, 2x - 5).

Area model — a rectangle split into regions by the terms of both factors; each region is one partial product, and the total area is the full product.

Partial product — one region's product: one term of the first factor times one term of the second. Two binomials always give four partial products before collecting.

Verify by substitution — replace the variable with the same number in the original factors and in the expanded product; the values must be equal.

Worked example

Expand (x + 3)(x + 5). Step 1: Draw the 2x2 area model in words: rows x and 3; columns x and 5. Four regions: x*x, x*5, 3*x, 3*5. Step 2: Partial products: x^2, 5x, 3x, 15. Step 3: Collect the like terms (the two x-regions): x^2 + 8x + 15. Step 4: This is exactly two-digit multiplication: 15 x 23 = (10 + 5)(20 + 3) = 200 + 30 + 100 + 15 = 345. Four partial products, then add. Algebra tiles are just base-ten blocks that keep going past 9.

Check yourself — explain it

Why do (x + 3)(x + 5) and 15 x 23 both produce four partial products?

Paired problem — example still visible

Expand (x + 2)(x + 7) -> x^2 + 9x + 14 (partials: x^2, 7x, 2x, 14; middle collects to 9x)

Faded completion — last step is yours

"(x - 4)(x + 6): partials x^2, 6x, -4x, ___; collected: x^2 + 2x - 24" -> -24 (-4 * 6)

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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Grade 10 — Mathematics 10C › C10U06 Polynomials: Multiply & Factor

L2 · Binomial times binomial