How do you simplify (x^2-6x+9)/(81-x^4)x2−6x+981−x4?

1 Answer
Shwetank Mauria
Jun 6, 2016

(x^2-6x+9)/(81-x^4)=(3-x)/((9+x^2)(3+x))x2−6x+981−x4=3−x(9+x2)(3+x)

Explanation:

To simplify (x^2-6x+9)/(81-x^4)x2−6x+981−x4, first factorize polynomials in numerator and denominator.

x^2-6x+9x2−6x+9 is complete square of the type (a-b)^2=a^2-2ab+b^2(a−b)2=a2−2ab+b2, as x^2-6x+9=(x)^2-2xx x xx 3 +3^2x2−6x+9=(x)2−2×x×3+32. Hence,

x^2-6x+9=(x-3)^2x2−6x+9=(x−3)2.

This can also be written as 9-6x+x^2=(3-x)^29−6x+x2=(3−x)2

For factorizing 81-x^481−x4, we use the identity (a^2-b^2)=(a+b)(a-b)(a2−b2)=(a+b)(a−b)

Hence 81-x^4=(9)^2-*x^2)^2=(9+x^2)(9+x^2)=(9+x^2)(3^2-x^2)81−x4=(9)2−⋅x2)2=(9+x2)(9+x2)=(9+x2)(32−x2)

= (9+x^2)(3+x)(3-x)(9+x2)(3+x)(3−x)

Hence (x^2-6x+9)/(81-x^4)=(3-x)^2/((9+x^2)(3+x)(3-x))x2−6x+981−x4=(3−x)2(9+x2)(3+x)(3−x)

= (3-x)/((9+x^2)(3+x))3−x(9+x2)(3+x)

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