Expanding Double Brackets

A binomial is a bracket with two terms, like (a + b). Expanding double brackets means multiplying two binomials together and removing the brackets: (a + b)(c + d) = ac + ad + bc + bd. Every term in the first bracket multiplies every term in the second, giving four products, like two pairs of friends who each say hello to the other pair.

Multiply each term in the first bracket by each term in the second, keeping the signs. For (2x − 1)(3x + 2): 2x × 3x = 6x², 2x × 2 = 4x, −1 × 3x = −3x and −1 × 2 = −2. Then collect like terms: 6x² + 4x − 3x − 2 = 6x² + x − 2. American sites call the order of the four products FOIL.

“Simplify” means collect the like terms after you expand. For (x + 3)(x + 2): the four products are x², 2x, 3x and 6, and the two x terms combine, so the answer is x² + 5x + 6. For (x − 2)(2x − 3): 2x² − 3x − 4x + 6 = 2x² − 7x + 6. Always finish by combining the middle terms.

Treat each minus as part of its term, so −1 is “plus negative 1”, then multiply with the signs. For (−3x + 2)(5 − y): −3x × 5 = −15x, −3x × (−y) = 3xy, 2 × 5 = 10 and 2 × (−y) = −2y. The result is −15x + 3xy + 10 − 2y, with no like terms to collect. Negative times negative is positive.

Expand two of the brackets first, then multiply the result by the third bracket, term by term. For (x + 1)(x + 2)(x + 3): (x + 1)(x + 2) = x² + 3x + 2, then (x² + 3x + 2)(x + 3) = x³ + 3x² + 3x² + 9x + 2x + 6, which collects to x³ + 6x² + 11x + 6. Lay the working out in a grid to avoid missing a product.

Because the two terms in a bracket are added, not multiplied: (a + b) means a plus b. You only multiply across the brackets, pairing each term of the first with each term of the second. In the video’s picture, a and b arrive together, so they do not need to say hello to each other, only to c and d.