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# General algebra

### Expanding brackets - using the Distributive Law

• In algebra, we often need to expand an expression, that is, to change it from a product to a sum, using the Distributive Law: $a(b+c) = ab + ac$
• Another type of expression that you may need to expand is $(x+3)(x+2)$
To do this we ensure that everything in the second bracket is multiplied by everything in the first.

For example:
$\newcommand{\eqncomment}{\scriptsize{\text{ #2}} } \newcommand{\ceqns}{ \begin{array}{rcll}} \newcommand{\ceqne}{\end{array}}$

$\ceqns &&(x+3)(x+2) \\ &=& x(x+2) + 3(x+2) & \eqncomment{0.4}{multiply the second bracket} \\ &&& \eqncomment{0.4}{into each term in the first,} \\ &=& x\times x+x\times 2+3\times x+3 \times 2 & \eqncomment{0.4}{expand the brackets,} \\ &=& x^2 +2x + 3x+ 6 \\ &=& x^2 + (2+3)x + 6 & \eqncomment{0.4}{group like terms,} \\ &=& x^2 + 5x +6\ \ceqne$

• Expand $$(2+x)(3-2x)(1+2x)$$:

\begin{eqnarray*}
&&(2+x)(3-2x)(1+2x)\\
&=& (2+x) \Big[3(1+2x)-2x(1+2x) \Big]\\
&=& (2+x) (3+6x-2x-4x^2) \\
&=& (2+x) (3+4x-4x^2) \\
&=& 2 (3+4x-4x^2) + x (3+4x-4x^2) \\
&=& 6 + 8x-8x^2 + 3x +4x^2 -4x^3 \\
&=& 6 + 11x - 4x^2 -4x^3\
\end{eqnarray*}