Breaking the Language Barrier Using Polynomials

The representative from the Factors is kind enough to remind you how to translate English into Factorian and vise-versa. She provides you with a convenient translation chart, as seen in Figure 1.

Figure 1. English-Factorian Translation Chart

    English      Factorian
       A             (x-1)(x+2)
       B            2(x-2)(x-3)
       C             (2x+3)(x-1)
       D             (x-3)(x+9)
       E             (x-4)(x+7)
       F             (x-4)(x-7)
       G             (x+3)(x-6)
       H             (x-5)(x+5)
       I             (x+5)(x-6)
       J             (x-6)(x-1)
       K             (x+1)(x-1)
       L             (x+3)(x-8)
       M             (x-8)(x+9)
       N             (x+4)(x-3)
       O             (x-2)(x-4)
       P             (x+7)(x-6)
       Q             (x-7)(x+6)
       R             (x-4)(x-6)
       S             (x-4)(7x-6)
       T            (3x+7)(9x+8)
       U             (x+1)(x+2)
       V            (4x-5)(x-1)
       W            (x+3)(x-7)
       X             (x+8)(x-9)
       Y            2(x-9)(x+9)
       Z             (x+5)(x+6)

So for example if we wish to write the letter A in Factorian, we would simply consult Figure 1, and see that A corresponds to the Factorian letter (x-1)(x+2).

The Factorian representative then informs you that if we wish to write the letter A in Polynomian, we must first convert A into its Factorian form (x-1)(x+2), and then expand

(x-1)(x+2)=x(x+2)-1(x+2)
              =x^2+2x-x-2
              =x^2+x-2,

yielding the Polynomian form of A. So A in Polynomian would appear as the letter x^2+x-2.

The Factorian representative then asks you to write the word HELLO in Polynomian, just to make sure you have understood her instructions. You then proceed:

H=(x-5)(x+5)
  =x(x+5)-5(x+5)
  =x^2+5x-5x-25
  =x^2-25

E=(x-4)(x+7)
  =x(x+7)-4(x+7)
  =x^2+7x-4x-28
  =x^2+3x-28

L=(x+3)(x-8)
  =x(x-8)+3(x-8)
  =x^2-8x+3x-24
  =x^2-5x-24

L=(x+3)(x-8)
  =x(x-8)+3(x-8)
  =x^2-8x+3x-24
  =x^2-5x-24

O=(x-2)(x-4)
  =x(x-4)-2(x-4)
  =x^2-4x-2x+8
  =x^2-6x+8

    English     Factorian     Polynomian
       H             (x-5)(x+5)          x^2-25
       E             (x-4)(x+7)       x^2+3x-28
       L             (x+3)(x-8)        x^2-5x-24
       L             (x+3)(x-8)        x^2-5x-24
       O             (x-2)(x-4)         x^2-6x+8

So HELLO=(x^2-25)(x^2+3x-28)(x^2-5x-24)(x^2-5x-24)(x^2-6x+8)

The Factorian representative approves, as you have correctly translated the word HELLO from English to Polynomian.

You are now ready to send a message to the Polynomes!!!