Algebra > Polynomials >
Algebra > Algebraic Identities >
Interactive Entries > Interactive Demonstrations >

Polynomial Identity

DOWNLOAD Mathematica Notebook

Polynomial identities involving sums and differences of like powers include

x^2-y^2=(x-y)(x+y)
(1)
x^3-y^3=(x-y)(x^2+xy+y^2)
(2)
x^3+y^3=(x+y)(x^2-xy+y^2)
(3)
x^4-y^4=(x-y)(x+y)(x^2+y^2)
(4)
x^5-y^5=(x-y)(x^4+x^3y+x^2y^2+xy^3+y^4)
(5)
x^5+y^5=(x+y)(x^4-x^3y+x^2y^2-xy^3+y^4)
(6)
x^6-y^6=(x-y)(x+y)(x^2+xy+y^2)(x^2-xy+y^2)
(7)
x^6+y^6=(x^2+y^2)(x^4-x^2y^2+y^4),
(8)

which are the polynomial versions of the so-called binomial numbers.

Further identities include

(x_1^2-Dy_1^2)(x_2^2-Dy_2^2)=(x_1x_2+Dy_1y_2)^2-D(x_1y_2+x_2y_1)^2
(9)
(x_1^2+Dy_1^2)(x_2^2+Dy_2^2)=(x_1x_2+/-Dy_1y_2)^2+D(x_1y_2∓x_2y_1)^2.
(10)

The identity

(X+Y+Z)^7-(X^7+Y^7+Z^7)=7(X+Y)(X+Z)(Y+Z)[(X^2+Y^2+Z^2+XY+XZ+YZ)^2+XYZ(X+Y+Z)]
(11)

was used by Lamé in his proof that Fermat's last theorem was true for n=7.

Wolfram Web Resources

Mathematica »

The #1 tool for creating Demonstrations and anything technical.

 Wolfram|Alpha »

Explore anything with the first computational knowledge engine.

 Wolfram Demonstrations Project »

Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.
Computerbasedmath.org »

Join the initiative for modernizing math education.

 Online Integral Calculator »

Solve integrals with Wolfram|Alpha.

 Step-by-step Solutions »

Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own.
Wolfram Problem Generator »

Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.

 Wolfram Education Portal »

Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more.

 Wolfram Language »

Knowledge-based programming for everyone.