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Colt1911 [192]
3 years ago
5

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

Mathematics
2 answers:
LenKa [72]3 years ago
7 0

Answer:

(n^2-6)/(n^2-2), n cannot be +-sqrt 5, +-sqrt 2

Step-by-step explanation:

djyliett [7]3 years ago
5 0
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}
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