The Art and Craft of Problem Solving

5.2 ALGEBRAIC MANIPULATION REVISITED 149

Many problems involve combinations of these fonnulas, along with basic strate­

gies (for example, wishful thinking), awareness of symmetry, and the valuable add

zero creatively tool.^5 Here is an example.

Example 5.2.9 Factor x^4 + 4 into two polynomials with real coefficients.

Solution: If it weren't for the requirement that the factors have real coefficients,

we could just treat x^4 + 4 as a difference of two squares (Fonnula (^5). 2 .6) and obtain
x^4 +4 = x^4 - ( - 4 ) = (�)^2 - (2i)^2 = (�+ 2i)(� -2i).
While we cannot use the difference-of-two-squares method directly, we should not
abandon it just yet, since the expression at hand contains two perfect squares. Un­
fortunately, it is not a difference of two perfect squares. But there are other possible
perfect squares, and our expression nearly contains them. Use wishful thinking to
make more perfect squares appear, by adding zero creatively.
x^4 +4 =x^4 +4x^2 +4- 4�.
This was the crux move, for now we have
x^4 +4� +4-4� = (�+2)^2 - (2x)^2 = (x^2 +2x+2)(x^2 - 2x+2).
This instructive example shows that you should always look for perfect squares, and
try to create them if they are not already there.

Manipulating Squares

Also well worth remembering is how to square a trinomial, not to mention more com­
plicated polynomials.

Know how to create and recognize perfect squares.

Toward this end, please leam the following fonnulas actively, not passively!

5.2.10 (x+ y + z)^2 = x^2 + y^2 + z^2 + 2.xy + 2xz + 2yz.

5.2. 11 (x+y+z+w)^2 = x^2 +y^2 +z^2 + w^2 + 2xy+ 2xz + 2xw+ 2yz+2yw+2zw.

5.2.12 Completing the square.

a^2 a^2

(

a

)

2 (a

)

2

�+ax=�+ax+

4

• 4
  = x+
  2"
  
  
  • 2"
    .
  
  
  
  

Ponder the completing-the-square fonnula above. One way to "discover" it is by
recognizing the perfect square that begins with x^2 + ax, and then adding zero cre­
atively. Another approach uses simple factoring, followed by an attempt to sym­
metrize the tenns, plus adding zero creatively:

� +ax =x(x+a) =

(

x+ � -�)

(

x+ � +�) =

(

x+ �r -

(�r·

(^5) The sister to the add zero creatively tool is the multiply cleverly by one tool.