Question

Show that 2x^6+12x^5+30x^4+60x^3+80x^2+30x+45=0 has no real roots

Answer 1

Answer:

First note

2x6+12x5+30x4+60x3+80x2+30x+45=2(x3+3x2)2+12(x2+52x)2+5(x+3)2.2x6+12x5+30x4+60x3+80x2+30x+45=2(x3+3x2)2+12(x2+52x)2+5(x+3)2.

Not only is this non-negative, but it could only be zero if

x3+3x2=x2+52x=x+3=0.x3+3x2=x2+52x=x+3=0.

The last condition simplifies to x=−3x=−3, which contradicts the second condition.

Step-by-step explanation:

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