After looking at the solutions to the first few problems, you will probably (hopefully) get the idea and be able to factor the
remaining problems yourself.
“Polynomials That Can Be Made the Difference of Two Squares”
a 2  b2  (a  b)(a  b)
Factor Each Of The Following:
1.
c 4  64  ???
4.
x 4  7 x 2  16  ???
5.
y 4  3 y 2  1  ???
6.
a 4  15a 2  9  ???
7.
9  16 x 2  4 x 4  ???
8.
25  26s 2  9s 4  ???
9.
a8  a 4  1  ???
10.
2.
4 y 4  1  ???
4r 4  17r 2  4  ???
11.
3.
z 4  5z 2  9  ???
25m4  11m2  4  ???
SOLUTIONS:
1.
c4  64  (c2  8)2  (4c)2  [(c2  8)  4c][( c2  8)  4c]  (c2  4c  8)(c2  4c  8)
2.
4 y 4  1  (2 y 2  1)2  (2 y )2  [(2 y 2  1)  2 y ][( 2 y 2  1)  2 y ]  (2 y 2  2 y  1)(2 y 2  2 y  1)
( z 4  z 3  3 z 2 )  (  z 3  z 2  3 z )  ( 3z 2  3 z  9 )
3.
z 4  5z 2  9  ( z 2  z  3)( z 2  z  3)
 ( z 4 )  ( z 3  z 3 )  (3z 2  z 2  3z 2 )  ( 3z  3z )  9
 z 4  5z 2  9
From this “immediate” solution, we can see that it is much easier to look at the problem as follows:
( z 2  3)2  z 4  6 z 2  9
z 2 from z 4  6 z 2  9 , and we may accomplish by the following:
[( z 2  3)  z ][( z 2  3)  z ]  ( z 2  3)2  z 2
So we want to subtract
Strange, but the first time I looked at these problems, I actually did not see the simple way to do them, but rather the
more complicated “immediate” way. Live and learn.
4.
x 4  7 x 2  16  ( x 2  4)2  x 2  [( x 2  4)  x][( x 2  4)  x]  ( x 2  x  4)( x 2  x  4)
5.
y 4  3 y 2  1  ( y 2  1)2  y 2  [( y 2  1)  y ][( y 2  1)  y ]  ( y 2  y  1)( y 2  y  1)
Since I converted the “immediate” forms to the simpler forms above, you get to do the conversions for the rest. Have fun!
6.
a 4  15a 2  9  (a 2  3a  3)(a 2  3a  3)
7.
9  16 x 2  4 x 4  (3  2 x  2 x 2 )(3  2 x  2 x 2 )
8.
25  26s 2  9s 4  (5  2s  3s 2 )(5  2s  3s 2 )
9.
a8  a 4  1  (a 4  a 2  1)(a 4  a 2  1)
10.
4r 4  17r 2  4  (2r 2  3r  2)(2r 2  3r  2)
11.
25m4  11m2  4  (5m2  3m  2)(5m2  3m  2)