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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:
\[
\newcommand{\eqncomment}[2]{\scriptsize{\text{ #2}} } 
\newcommand{\ceqns}{ \begin{array}{rcll}}
\newcommand{\ceqne}{\end{array}}
\]

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

  • 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*}