Algebra

Triple brackets

Multiply in stages. Collect matching powers.

Predict, test and explain

Step 1 of 3

Predict

What is the coefficient of x² in (x + 1)(x + 2)(x + 3)?

(x+1)(x+2)(x+3)(x + 1)(x + 2)(x + 3)
(x+1)(x+2)(x+3)(x + 1)(x + 2)(x + 3)
Multiply two brackets first.\text{Multiply two brackets first.}
Then multiply every term by the third.\text{Then multiply every term by the third.}

Scroll the diagram sideways to see every label.

step 1\text{step }1

InvestigateWhere do all three square terms come from?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Expand and simplify.

(x−1)(x+1)(x+2)(x-1)(x+1)(x+2)

Hint

The first two brackets form a difference of two squares.

Reveal answer

Expand the first two brackets, then multiply both resulting terms by the third bracket.

(x−1)(x+1)(x+2)=x3+2x2−x−2(x-1)(x+1)(x+2)=x^3+2x^2-x-2

Build an explanation

Hint 1

Start by multiplying two brackets.

Hint 2

Multiply every term of that result by each term of the third bracket.

Hint 3

Collect cubic, square, linear and constant terms separately.

Worked example

This example uses fixed values, separate from the diagram controls.

Expand (x + 1)(x + 2)(x + 3).

  1. The first two brackets give x² + 3x + 2.
  2. Multiply by x + 3 to get x³ + 3x² + 3x² + 9x + 2x + 6.
  3. Collect terms with the same power.

Answer: x³ + 6x² + 11x + 6.

Connect this idea

Watch out: x² and x are different powers and cannot be combined into one like term.