The Function x^2 + xy + y^2 – Part 1

At some point in high school, most students learn how to factor a difference of cubes.  The general formula for factoring a difference of cubes is x³ – y³ = (x – y)(x² + xy + y²).  So to factor a³ – 64, we would let x = a and y = 4, which means a³ – 64 = a³ – 4³ = (a – 4)(a² + a∙4 + 4²) = (a – 4)(a² + 4a + 16).

Many students erroneously try to factor the x² + xy + y² part, but it can’t be done with real numbers. Other quadratics can be factored, such as x² + 10x + 16.  (To factor x² + 10x + 16, we would need to numbers p and q that multiply to 16 and add up to 10.  These two numbers are p = 2 and q = 8, so x² + 10x + 16 = (x + 2)(x + 8).)  But to factor x² + xy + y², we would need two numbers to multiply to y² and one number to add up to y.  In other words, two numbers p and q such that pq = y² and p + q = y.  This means that q = y – p, and p(y – p) = y² or py – p² = y² or p² – py + y² = 0.  The discriminant of this quadratic equation is b² – 4ac = (-y)² – 4∙1∙y² = -3y² which is always negative, which means there are always two imaginary answers and zero real answers, which makes x² + xy + y² not factorable with integers.

But even though it can’t be factored, x² + xy + y² has some amazing properties.  When x = 1 and y = 1, which we will denote as (1, 1), 1² + 1∙1 + 1² = 3.  Continuing in this fashion, (1, 2) = 7, (1, 3) = 13, (1, 4) = 21, and (1, 5) = 31, etc. Let’s investigate x² + xy + y² by making a number chart.

	1	2	3	4	5	6	7	8	9	10
1	3	7	13	21	31	43	57	73	91	111
2	7	12	19	28	39	52	67	84	103	124
3	13	19	27	37	49	63	79	97	117	139
4	21	28	37	48	61	76	93	112	133	156
5	31	39	49	61	75	91	109	129	151	175
6	43	52	63	76	91	108	127	148	171	196
7	57	67	79	93	109	127	147	169	193	219
8	73	84	97	112	129	148	169	192	217	244
9	91	103	117	133	151	171	193	217	243	271
10	111	124	139	156	175	196	219	244	271	300

Now that we have a made a table of values, we can make several interesting observations about x² + xy + y².  Here are ten of those observations.

1)      x² + xy + y² has the commutative property, but not the associative or distributive properties.

The function x² + xy + y² is commutative, because (x, y) = x² + xy + y² = y² + yx + x² = (y, x).  However, x² + xy + y² is not associative or distributive.  We can show that it is not associative by giving a counter-example of ((2, 3), 4) ≠ (2, (3, 4)), because ((2, 3), 4) = (19, 4) = 453 but (2, (3, 4)) = (2, 37) = 1447.  We can also show that it is not distributive by giving a counter-example of (2, (3 + 4)) ≠ (2, 3) + (2, 4), because (2, (3 + 4)) = (2, 7) = 67 but (2, 3) + (2, 4) = 19 + 28 = 47.  The expression x² + xy + y² also has an identity but it is not apparent from the table of values since it is usually an imaginary number.  If there is an identity e so that (x, e) = x, then x² + xe + e² = x, or e² + xe + x² – x = 0.  Using the quadratic equation, e =  ½(-x ± Ö(x² – 4(x² – x))) = -½x ± ½Ö(-3x² + 4x).  This means that the identity for 2 is -½(2) ± ½Ö(-3(2)² + 4(2)) = -1 ± i.