Expanding three brackets - Higher

To expand three brackets, expand and simplify two of the brackets then multiply the resulting expression by the third bracket.

Example

  • expand and simplify \((x + 3)(x – 4)(2x + 5)\)
  • first, expand and simplify \( (x + 3)(x – 4)\) to give \((x + 3)(x – 4) = x^2 + 3x – 4x - 12 = x^2 – x – 12\)
  • then, expand and simplify \((2x + 5)( x^2 – x – 12)\)

One way to organise this is using a grid:

A grid that has expanded and simplified (x + 3)(x  - 4)(2x + 5)

Finally, collect like terms: \(2x^3 + 5x^2 – 2x^2 – 5x – 24x – 60 = 2x^3 + 3x^2 – 29x – 60 \)

So, \((x + 3)(x – 4)(2x + 5) = 2x^3 + 3x^2 – 29x – 60\)

Question

Expand and simplify \((x – 2)(x + 6)(x – 3)\)

First, expand and simplify \((x – 2)(x + 6) \) to give \(x^2 – 2x + 6x -12 = x^2 +4x – 12\).

Then, multiply this answer by \((x – 3) \) to give \(x^3 + 4x^2 – 12x – 3x^2 – 12x + 36\) which simplifies to give \(x^3 + x^2 – 24x + 36 \).