THIS USER ASKED πŸ‘‡

Simplify the rational expression. state any restrictions on the variable. n4-10n^2+24/n^4-9n^2+18

THIS IS THE BEST ANSWER πŸ‘‡

ANSWER

 frac {n ^ 4-10n ^ 2 + 24} {n ^ 4-9n ^ 2 + 18} =  frac {(n-2) (n + 2)} {(n-  sqrt {3}) ( n +  sqrt {3})}

For

n  ne  pm  sqrt {3}

EXPLANATION

We have  frac {n ^ 4-10n ^ 2 + 24} {n ^ 4-9n ^ 2 + 18}

This is very easy to simplify. Let’s look at both expressions from a trinomial quadratic perspective.

We rewrote the rational expression to be found;

 frac {n ^ 4-10n ^ 2 + 24} {n ^ 4-9n ^ 2 + 18} =  frac {(n ^ 2) ^ 2-10 (n ^ 2) +24} {(n ^ 2 ) ^ 2-9 (n ^ 2) +18}

We can now see that both the numerator and the denominator are square trinomials n ^ 2.

We divide the average terms as follows;

 frac {n ^ 4-10n ^ 2 + 24} {n ^ 4-9n ^ 2 + 18} =  frac {(n ^ 2) ^ 2-6n ^ 2-4n ^ 2 + 24} {(n ^ 2) ^ 2-6n ^ 2-3n ^ 2 + 18}

 frac {n ^ 4-10n ^ 2 + 24} {n ^ 4-9n ^ 2 + 18} =  frac {n ^ 2 (n ^ 2-6) -4 (n ^ 2-6)} {n ^ 2 (n ^ 2-6) -3 (n ^ 2-6)}

We get more;

 frac {n ^ 4-10n ^ 2 + 24} {n ^ 4-9n ^ 2 + 18} =  frac {(n ^ 2-6) (n ^ 2-4)} {(n ^ 2-6 ) (n ^ 2-3)}

We now dismiss common factors;

 frac {n ^ 4-10n ^ 2 + 24} {n ^ 4-9n ^ 2 + 18} =  frac {(n ^ 2-4)} {(n ^ 2-3)}

 frac {n ^ 4-10n ^ 2 + 24} {n ^ 4-9n ^ 2 + 18} =  frac {(n-2) (n + 2)} {(n-  sqrt {3}) ( n +  sqrt {3})}

For

n  ne  pm  sqrt {3}