Question 1184738
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Can these be factored using the methods (GCF, squares, cubes)
(a) a4 + 1
(b) a4 - 64
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<pre>
(b)  a^4 - 64  can be factored this classic way

         {{{a^4-64}}} = {{{(a^2-8)*(a^2+8)}}}

    over integer numbers.


    It can be factored further as

         {{{a^4-64}}} = {{{(a^2-8)*(a^2+8)}}} = {{{(a-sqrt(8))*(a+sqrt(8))*(a^2+8)}}} = {{{(a-2sqrt(2))*(a+2sqrt(2))*(a^2+8)}}}

    but just over real numbers, not over integer or rational numbers.


    The factor {{{a^2+8}}}  can be factored further over COMPLEX numbers.




(a)  {{{a^4 + 1}}}  can be factored in this way

         {{{a^4+1}}} = {{{(a^4 + 2a^2 + 1)}}} - {{{2a^2}}} = {{{(a^2+1)^2}}} - {{{(sqrt(2)*a)^2}}} = 

               = {{{(a^2-sqrt(2)*a +1)*(a^2+sqrt(2)*a+1)}}}.


     This factoring works over real numbers and includes the irrational number {{{sqrt(2)}}}.

     It does not work over integer or rational numbers.
</pre>

Solved, answered and explained.


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See the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Advanced-factoring.lesson>Advanced factoring</A>

in this site.