If $a$ and $b$ are integers such that $x^2 -x-1$ is a factor of $ax^3 + bx^2 +1$ , then find b?

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Answer 1 · 2017-06-07T15:20:14

$b=-2$

If $x^2-x-1$ is a factor of $a x^3 + b x^2 + 1$ then

$a x^3 + b x^2 + 1=(c x + d) (x^2 - x - 1)$ then grouping coefficients

$(a-c)x^3+(b+c-d)x^2+(c+d)x+d+1 = 0$

This must be true for all $x$ so solving

${(d+1=0),(c+d=0),(b+c-d=0),(a-c=0):}$

we have

$a=1,b=-2,c=1,d=-1$ so

$b=-2$

If aa and bb are integers such that x^2 -x-1x^2 -x-1 is a factor of ax^3 + bx^2 +1ax^3 + bx^2 +1, then find b?