Simplify the rational expression n^4-11n^2+30/n^4-7n^2+10

Asked by maham237 @ 17/12/2020 in Mathematics viewed by 345 People

Simplify the rational expression. State any restrictions on the variable n^4-11n^2+30/ n^4-7n^2+10

Answered by maham237 @ 17/12/2020

To factor both numerator and denominator in this rational expression we are going to substitute $n^{2}$ with $x$ ; so $n^{2} =x$ and $n ^{4} = x^{2}$ . This way we can rewrite the expression as follows:
$\frac{n^{4}-11n^{2} +30 }{n^{2}-7n^{2} +10 } = \frac{ x^{2} -11x+30}{ x^{2} -7x+10}$
Now we have two much easier to factor expressions of the form $a x^{2} +bx+c$ . For the numerator we need to find two numbers whose product is $c$ (30) and its sum $b$ (-11); those numbers are -5 and -6. $(-5)(-6)=30$ and $-5-6=-11$ .
Similarly, for the denominator those numbers are -2 and -5. $(-2)(-5)=10$ and $-2-5=-7$ . Now we can factor both numerator and denominator:
$\frac{ x^{2} -11x+30}{ x^{2} -7x+10} = \frac{(x-6)(x-5)}{(x-2)(x-5)}$
Notice that we have $(x-5)$ in both numerator and denominator, so we can cancel those out:
$\frac{x-6}{x-2}$
But remember than $x= n^{2}$ , so lets replace that to get back to our original variable:
$\frac{n^{2}-6 }{n^{2}-2 }$
Last but not least, the denominator of rational expression can't be zero, so the only restriction in the variable is $n^{2} -2 \neq 0$
$n^{2} \neq 2$
$n \neq +or- \sqrt{2}$

Simplify the rational expression n^4-11n^2+30/n^4-7n^2+10

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