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Factoring Binomials: Difference of Two Squares

Updated on September 25, 2011

Factoring Binomials : Difference of Two Squares

Difference of Two Squares

X ^2 – Y^2=(X +Y ) ( X – Y)

The difference of two squares of two numbers is equal to the productof the sum and difference of the two numbers.

Example One :Factor64X^2-81^2

First let us inspect if the given problem is a difference of two squares. The numerical coefficients must be perfect squares. It means that their square root is an exact whole number. For literal coefficients, their exponents must be divisible by two. Only by meeting these conditions it can be factored as difference of two squares.

64 is a perfect square and its square root is 8. 81 is a perfect square and its square root is 9.

64^2 – 81^2 can be rewritten as ( 8X)^2-(9Y)^2therefore its factors are(8x + 9Y) (8X – 9Y).

Example Two :FactorX^8 – Y^8

X^8 – y^8=( x^4-)^2-(Y^4)^2

=( X^4 + Y^4) (X^4 – Y ^4)

=(X^4 + Y^4 ) (X^2 + Y^2) (X^2 – Y^2)

=(X^4 + Y^4 ) (X^2 + Y^2 ) (X +Y ) ( X – Y)

Example Three : Factor(a + 3b ) ^2 – 16c^4

(a + 3b)^2- 16 c^4=(a + 3b)^2 – (4c^2)^2

=(a + 3b+ 4c^2) (a + 3b – 4c^2)

Example Four : Factor49a^10b^8- 100c^6

49a^10b^8- 100c^6=( 7a^5b^4)^2-(10c^3)^2

=(7a^5b^4+- 10 c^3)(7a^5b^4- 10 c^3)

Example Five : Factor81X^12-256^Y^8

81X^12 – 256Y^8=(9X^6)^2-(16Y^4)

=(9X^6 + 16Y^4) (9X^6 – 16Y^4)

=(9X^6+ 16 Y^4) (3X^3 + 4Y^2) (3X^3 – 4Y^2)


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    • cristina327 profile imageAUTHOR

      Maria Cristina Aquino Santander 

      7 years ago from Manila

      HI ronhi thank you for dropping by. Regards.

    • cristina327 profile imageAUTHOR

      Maria Cristina Aquino Santander 

      7 years ago from Manila

      Thank you lord de cross for appreciating this hub. Blessings to you always and best regards.

    • ronhi profile image


      7 years ago from Kenya

      where were you when i was failing maths in school?

    • Lord De Cross profile image

      Joseph De Cross 

      7 years ago from New York

      Excellent and well explained Cristina. Seen so many kids struggling with Algebra thru the years.Useful and voted up!



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