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Integral

i have a math problem : Question :

consider the integral expression in x

P=x^3+x^2+ax+1

where a is a rational number. at a = (...........) the value of P is a rational number for any x which satisfies the equation x^2+2x-2=0 , and in this case the value of P is (.........)

3 years, 11 months ago

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If $$x^2+2x-2=0$$, then $$x^2=-2x+2$$. Also, $$(x^2+2x-2)(x-2)=x^3-6x+4=0$$ so that $$x^3=6x-4$$.

If you substitute these in to $$P$$, you get the expression in terms of only the first degree of $$x$$: $$P=x^3+x^2+ax+1=(6x-4)+(-2x+2)+ax+1=(a+4)x-1$$.

You can see that by making $$a=-4$$, $$P$$ will be rational (specifically, $$P$$ will equal $$-1$$). Note that no other rational value of $$a$$ works because $$(a+4)x-1$$ would be a nonzero rational number ($$a+4$$) times an irrational number ($$x$$) minus a rational number ($$-1$$), which would be irrational. · 3 years, 11 months ago

from what (x-2) ? · 3 years, 11 months ago

Sorry that I didn't explain that. I chose to multiply $$x^2+2x-2$$ by $$x-2$$ because that introduces an $$x^3$$ term (so that you can make the substitution) and because the result doesn't have an $$x^2$$ term (which would get in the way in the substitution, although even if there were an $$x^2$$ term you could use $$x^2=-2x+2$$ to reduce the second degree to first degree). · 3 years, 11 months ago

oh, i understand. i have other some math problem

this problem :

consider two conditions x^2-3x-10<0 and |x-2|<a ona real number x, where a is a positive real number.

(1) necessary and sufficient condition for x^2-3x-10<0 that (......)<x<(......)

(2) the range of values of a such that |x-2|<a is necessary condition for x^2-3x-10 is (......)

(3) the range of values of a such that |x-2|<a is sufficient condition for x^2-3x-10 is (......) · 3 years, 11 months ago