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Factor the following differences of squares.

  1. \(\frac{1}{{25}}x^2 - 9\)

    \[\frac{1}{{25}}x^2 - 9 = \left( {\frac{1}{5}x + 3} \right)\left( {\frac{1}{5}x - 3} \right)\]


  2. \(64x^2 - y^6 \)

    \(64x^2 - y^6  = \left( {8x + y^3 } \right)\left( {8x - y^3 } \right)\)

  3. \(\left( {x^2 - 2y^2 } \right)^2 - 4y^4\)
    \(\begin{align}
     \left( {x^2 - 2y^2} \right)^2 - 4y^4 &= \left[ {\left( {x^2 - 2y^2} \right) + 2y^2} \right]\left[ {\left( {x^2 - 2y^2} \right) - 2y^2} \right] \\
      &= x^2 \left( {x^2 - 4y^2} \right) \\
      &= x^2 \left( {x - 2y} \right)\left( {x + 2y} \right) \\
     \end{align}\)

    Notice that the second factor can be factored further because it is itself a difference of squares!


 For further information about Factoring Polynomials see pp. 218 to 222 of Pre-Calculus 11.