Which of the following is [tex]x^4 + y^4[/tex] identically equal to?
A. \((x^2 + y^2)(x + y)(x - y)\)
B. \((x^2 + xy + y^2)(x^2 - xy + y^2)\)
C. \((x^2 + \sqrt{2}xy + y^2)(x^2 - \sqrt{2}xy + y^2)\)
D. \((x^2 + \sqrt{3}xy + y^2)(x^2 - \sqrt{3}xy + y^2)\)
E. \((x + y)^2(x - y)^2\)
Can someone explain to me how to solve it step by step?
Answer (3)
x 4 + y 4 x 4 = ( x 2 ) 2 , y 4 = ( y 2 ) 2 x 4 − y 4 = ( x 2 ) 2 − ( y 2 ) 2 = ( x 2 + y 2 ) ( x 2 − y 2 ) n o w yo u c an f a c t or t h e seco n d a g ain ( h o p e f u ll y yo u see t ha t i t ′ s a d i ff ere n ce o f s q u a res ) : x 4 − y 4 = ( x 2 + y 2 ) ( x + y ) ( x − y ) A n s w er : ( A ) ( x 2 + y 2 ) ( x + y ) ( x − y ) a 2 − b 2 = ( a − b ) ( a + b )
The correct option is (B) (x² + xy + y²)(x² - xy + y²). Upon simplifying each given expression, only option (B) simplifies to x⁴ + y⁴ without any additional or different terms, making it the correct choice.
To determine which of the given options is identically equal to x⁴ + y⁴, let's check them by expanding and simplifying where necessary.
Option (A): (x² + y²)(x + y)(x - y) expands to x²(x² - y²) + y²(x² - y²), which simplifies to x⁴ - x²y² + y⁴ - x²y². As you can see, this does not simplify to x⁴ + y⁴ because of the negative terms - x²y².
Option (B): (x² + xy + y²)(x² - xy + y²) expands to x⁴ + y⁴ with the cross terms xy canceling each other out, hence this is the correct choice.
The other options can be checked similarly, but they will include additional terms with xy, square roots, or different powers that do not simplify to the original expression x⁴ + y⁴.
Therefore, the correct answer is option (B) (x² + xy + y²)(x² - xy + y²).
The expression x 4 + y 4 can be factored into ( x 2 + x y + y 2 ) ( x 2 − x y + y 2 ) , which corresponds to Option B. Other options either do not factor correctly or represent different polynomial identities. Thus, Option B is the correct answer.
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