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Analyze the product x4 + x2 + 1? |
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Answer» Answer: To FIND this, first notice that x 4 + x 2 + 1 > 0 for all (real) values of x . So there are no linear factors, only quadratic ones. x 4 + x 2 + 1 = ( a x 2 + b x + c ) ( d x 2 + e x + f )
Without bothering to multiply this out fully just yet, notice that the coefficient of x 4 gives us a d = 1 . We might as WELL let a = 1 and d = 1 . ... = ( x 2 + b x + c ) ( x 2 + e x + f )
NEXT, the coefficient of x 3 gives us b + e = 0 , so e = − b . ... = ( x 2 + b x + c ) ( x 2 − b x + f )
The constant term gives us c f = 1 , so either c = f = 1 or c = f = − 1 . Let's try c = f = 1 . ... = ( x 2 + b x + 1 ) ( x 2 − b x + 1 )
Note that the coefficient of x will VANISH nicely when these are multiplied out. Finally notice that the coefficient of x 2 is ( 1 − b 2 + 1 ) = 2 − b 2 , giving us 1 = 2 − b 2 , thus b 2 = 1 , so b = 1 or b = − 1 . |
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