已知x^2-2x=5,求x^4-3x^3-12x^2-31x-15

Question

已知x^2-2x=5,求x^4-3x^3-12x^2-31x-15

Answer 1

x^4-3x³-12x²-31x-15
=x^4-4x³+4x²+x³-16x²-31x-15
=(x^4-4x³+4x²)+x³-2x²-14x²-31x-15
=(x^4-4x³+4x²)+(x³-2x²)-14x²+28x-59x-15
=(x^4-4x³+4x²)+(x³-2x²)-14(x²-2x)-59x-15
=(x²-2x)²+x(x²-2x)-14(x²-2x)-59x-15
=5²+5x-14×5-59x-15
=-54x-60
由x²-2x=5
得x²-2x+1=6
(x+1)²=6
x+1=±√6
x1=√6-1 ,x2=-√6-1
①当x=√6-1时,原式=-54(√6-1)-60=-54√6-6
②当x=-√6-1时,原式=-54(-√6-1)-60=54√6-6