Double and triple brackets

In a nutshell

Double brackets consist of two binomial expressions multiplied together, e.g. $(x+1)(y-3)$ , whereas triple brackets consist of three binomial expressions multiplied together, e.g. $(x-2)(x+6)(x+9)$ . The principle behind multiplying out double brackets can be extended to multiplying out triple brackets, using the FOIL method, or a multiplication grid.

Recap on multiplying double brackets

You can expand two pairs of brackets by multiplying each term in the first set of brackets by each term in he second set. This can be done by using a multiplication grid.

Example 1

Expand the brackets $(x+1)(2x-3)$

Fill in the multiplication grid

$\times$	$\boldsymbol{x}$	$\bold{+1}$
$\boldsymbol{2x}$	$2x^2$	$+2x$
$\bold{-3}$	$-3x$	$-3$

Add and simplify the results

$\begin {aligned} (x+1)(2x-3) &= 2x^2 +2x-3x-3 \\&= \underline{2x^2-x-3}\end {aligned}$

Note: when there are just two binomial expressions, you can also use the FOIL method, as discussed in the previous lesson (multiply the First terms, the Outer terms, Inside terms and the Last terms).

Expanding Triple Brackets

To expand three brackets, first expand two of the brackets, then use the result to expand with the final pair. Use a multiplication grid to help.

Example 2

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

Answer

Take the first two brackets, $(x+1)(x+2)$ and multiply out using a grid.

$\times$	$\boldsymbol{x}$	$\bold{+1}$
$\boldsymbol{x}$	$x^2$	$+x$
$\bold{+2}$	$+2x$	$+2$

Add the terms and simplify

$\begin {aligned} (x+1)(x+2) &= x^2 +x+ 2x+ 2 \\ &= x^2 + 3x +2 \end {aligned}$

Substitute back into the original question

$(x+1)(x+2)(x+3) = (x^2 + 3x+2)(x+3)$

Use another multiplication grid to multiply out these two brackets

$\times$	$\boldsymbol{x^2}$	$\boldsymbol{+3x}$	$\bold{+2}$
$\boldsymbol{x}$	$x^3$	$+3x^2$	$+2x$
$\bold{+3}$	$+3x^2$	$+9x$	$+6$

Add the terms and simplify

$\begin {aligned}(x+1)(x+2)(x+3) &= (x^2+3x+2)(x+3) \\&= x^3+3x^2+2x+3x^2+9x+6 \\&= \underline{x^3 + 6x^2+11x+6} \end {aligned}$

With practice, it is possible to multiply out without using the grid, using the FOIL method to multiply the first two brackets, and then multiply each term in the result with every term in the third bracket.