- Education and Science
Factoring Binomials: Difference of Two Squares
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)