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 (.........)

please help me in answering the question on its points thak you for your help

Easy Math Editor

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Comments

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.

Ryoma Canastra - 8 years, 1 month ago

from what (x-2) ?

Fadlan Semeion - 8 years, 1 month 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).

Ryoma Canastra - 8 years, 1 month ago

@Ryoma Canastra – 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 (......)

Fadlan Semeion - 8 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$