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Polynomial Identity

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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.

See also

Binomial Number, Euler Four-Square Identity, Gauss's Polynomial Identity, Liouville Polynomial Identity, Perfect Cubic Polynomial, Polynomial

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Cite this as:

Weisstein, Eric W. "Polynomial Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PolynomialIdentity.html

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