L3 · Polynomial times polynomial (the trinomial ceiling)

Learning goal

“I can multiply a binomial by a trinomial (or a trinomial by a binomial) by distributing every term, keep the six partial products organized, and verify the result by substitution.”

Trinomial — a polynomial with exactly three terms (e.g. x^2 + 3x + 1).

Partial product — (review from L2) one term-by-term product; a binomial x a trinomial gives 2 x 3 = 6 partial products.

Degree — the highest power of the variable in a polynomial; a degree-1 factor times a degree-2 factor gives degree 3.

Collect like terms — (review) combine partials with identical variable parts before writing the final descending-powers answer.

Worked example

Expand (x + 2)(x^2 + 3x + 1). Step 1: The area model is now a 2x3 grid: rows x and 2; columns x^2, 3x, 1. Six regions. Step 2: Distribute x first: x*x^2 = x^3, x*3x = 3x^2, x*1 = x. Step 3: Distribute 2 second: 2*x^2 = 2x^2, 2*3x = 6x, 2*1 = 2. Step 4: Collect by power: x^3; 3x^2 + 2x^2 = 5x^2; x + 6x = 7x; 2. Answer: x^3 + 5x^2 + 7x + 2. The routine that survives: one distributor at a time, one full row of the grid at a time.

Check yourself — explain it

Why does the product have degree 3 even though no factor does?

Paired problem — example still visible

Expand (x + 1)(x^2 + 2x + 4) -> x^3 + 3x^2 + 6x + 4 (x-row: x^3, 2x^2, 4x; 1-row: x^2, 2x, 4; collect: 3x^2, 6x)

Faded completion — last step is yours

"(x - 3)(x^2 + x + 2): partials x^3, x^2, 2x, -3x^2, -3x, -6; collected: x^3 ___ - x - 6" -> - 2x^2 (x^2 - 3x^2 = -2x^2)

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

L3 · Polynomial times polynomial (the trinomial ceiling)