Question

（ax+by)^2+（ay-bx)^2和a^4+a^2× b^2+b^4因式分解

Answer 1

(ax+by)^2+（ay-bx)^2
=a²x²＋2abxy＋b²y²＋a²y²－2abxy＋b²x²
=a²x²＋b²y²＋a²y²＋b²x²
=(a²x²＋a²y²)＋(b²y²＋b²x²)
=a²(x²＋y²)＋b²(x²＋y²)
=(a²＋b²)(x²＋y²)

a⁴＋a²b²＋b⁴
=(a⁴＋2a²b²＋b⁴)－a²b²
=(a²＋b²)²－(ab)²
=(a²＋ab＋b²)(a²－ab＋b²)

Answer 2

（ax+by)^2+（ay-bx)^2
解：=a²x²+2abxy+b²y² +a²y²-2abxy+b²x²
=a²x²+b²y² +a²y²+b²x²
=(a²x²+a²y²)+( b²x²+b²y² )
=a²(x²+y²)+b²(x²+y²)
=(x²+y²)(a²+b²)

a^4+a^2× b^2+b^4
=a^4+2a^2× b^2+b^4-a²b²
=(a^4+2a² b²+b^4)-a²b²
=(a²+b²)²-(ab)²
=(a²+b²+ab)(a²+b²-ab)

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Answer 3

（ax+by)^2+（ay-bx)^2
=a^2x^2+2abxy+b^2y^2+a^2y^2-2abxy+b^2x^2
=a^2x^2+b^2y^2+a^2y^2+b^2x^2+b^2y^2
=a^2(x^2+y^)+b^2(x^2+y^2)
=(a^2+b^2)(x^2+y^2)

a^4+a^2× b^2+b^4
=a^4+2a^2× b^2+b^4-a^2b^2
=(a^2+b^2)^2-a^2b^2
=(a^2+b^2+ab)(a^2+b^2-ab)