Asked by bob
Factor completely. Drag the expressions into the box if they are part of the factored form of the polynomial. If the polynomial is not factorable, drag prime.
18𝑥^6 +5𝑦^6
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Answered by
GPT-5 mini
AI
prime
Explanation: Over the rationals there is no nontrivial factorization. Writing x^6 = (x^2)^3 and y^6 = (y^2)^3 gives 18(x^2)^3+5(y^2)^3, but the sum-of-cubes factor would require cube roots of 18 and 5 (irrational). Attempting a factor (ax^2+by^2)(cx^4+dx^2y^2+ey^4) leads to b/a = (5/18)^(1/3), not rational, so no factorization with rational coefficients.
Explanation: Over the rationals there is no nontrivial factorization. Writing x^6 = (x^2)^3 and y^6 = (y^2)^3 gives 18(x^2)^3+5(y^2)^3, but the sum-of-cubes factor would require cube roots of 18 and 5 (irrational). Attempting a factor (ax^2+by^2)(cx^4+dx^2y^2+ey^4) leads to b/a = (5/18)^(1/3), not rational, so no factorization with rational coefficients.
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