L1 · Multiplying a polynomial by a monomial

Learning goal

“I can multiply a monomial by any polynomial - term by term - and check my product by substituting a number for the variable.”

Monomial — a single term: a number, a variable, or a product of numbers and variables (e.g. 5, x, -3x^2, 2xy).

Term — one piece of a polynomial, separated by + or - signs (3x^2 - 2x + 7 has three terms).

Distributive property — a multiplier outside brackets multiplies EVERY term inside: a(b + c) = ab + ac.

Like terms — terms with the same variables raised to the same powers; only like terms can be collected (2x^2 + 3x^2 = 5x^2, but x^2 and x never merge).

Worked example

Expand 3x(2x + 5). Step 1: Picture the area model: a rectangle of height 3x, split into two regions of widths 2x and 5. Two regions, so the product must have two terms. Step 2: Distribute: 3x * 2x = 6x^2 (coefficients 3 * 2 = 6; x * x = x^2) and 3x * 5 = 15x. Step 3: Write in descending powers: 6x^2 + 15x. Step 4 (the habit starts now): verify by substituting x = 2. Original: 3(2)(2*2 + 5) = 6 * 9 = 54. Product: 6(4) + 15(2) = 24 + 30 = 54.

Check yourself — explain it

Why does 3x(2x + 5) have TWO terms in its answer?

Paired problem — example still visible

Expand 4a(a - 3) -> 4a^2 - 12a (4a * a = 4a^2; 4a * (-3) = -12a)

Faded completion — last step is yours

"-2x(4x + 1) = -8x^2 + ___" -> -2x (-2x * 1 = -2x: the multiplier still applies to the plain 1, and it stays negative)

Ready to practise?

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

Start practice →Take the unit quiz

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

L1 · Multiplying a polynomial by a monomial