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Expanding expressions

Algebraic expressions can sometimes be complicated!

$$$ (3\Tred{x} - 2)(2\Tred{x}^2 + \Tred{x} - 1) + 2$$$

The expansion of an algebraic expression removes all the brackets and collects like terms. It is then quicker to evaluate the expression.

  • Remove the brackets one by one by to find the products: \begin{align*} (\Tred{a}+\Tblue{b})(c+d) &= \Tred{a}(c+d) + \Tblue{b}(c+d) \\ &= \Tred{a}c + \Tred{a}d + \Tblue{b}c + \Tblue{b}d \end{align*}
  • Get the signs right: \begin{align*} \Tred{\mathbf{+}}\times \Tred{\mathbf{+}} = \Tred{\mathbf{+}} &\qquad \Tred{\mathbf{+}}\times \Tblue{\mathbf{-}} = \Torange{\mathbf{-}}\\ \Tblue{\mathbf{-}}\times \Tblue{\mathbf{-}} = \Tred{\mathbf{+}} &\qquad \Tblue{\mathbf{-}}\times \Tred{\mathbf{+}} = \Tblue{\mathbf{-}} \end{align*}
  • Rewrite products of variables using powers: $$$ a\times a = a^2,\quad a^2\times a = a\times a^2 = a^3$$$
  • Collect all the like terms.
$$$\Tblue{2}(y-1) - 1= \Tblue{2}y + \Tblue{2}(-1) - 1 = 2y - 2 - 1 = 2y -3$$$ \begin{align*} 2(\Tred{t}+1)(\Tred{t}-2) - 1 &= (2\Tred{t}+2)(\Tred{t}-2) - 1 \\ &= 2\Tred{t}\times \Tred{t} + 2\Tred{t} - 4\Tred{t} - 4 -1\\ &= 2\Tviolet{t^2} + (2-4)\Tred{t} + (-4-1) \\ &= 2\Tviolet{t^2} -2\Tred{t} -5 \\ \end{align*}