Simplify the rational expression. State any restrictions on the variable.
(n^4 - 11n^2 +30)/(n^4 - 7n^2 + 10)
[(n - 6)(n - 5)(n + 1)(n + 2)] / [(n - 5)(n - 2)(n - 1)(n - 1)]
Restrictions: n cannot equal 5, 2, or 1.
the options are
A n^2 - 6/n^2 - 2; n ≠ 5, n ≠ 2
B -(n^2 - 6)/n^2 - 2; n ≠ ± sqrt5, n ≠ ± sqrt2
C n^2 - 6/n^2 - 2; n ≠ ± sqrt5, n ≠ ± sqrt2
D n^2 - 6/n^2 - 2; n ≠ 5, n ≠ -2
Let's factor the numerator and denominator completely:
Numerator: n^4 - 11n^2 + 30 = (n^2 - 6)(n^2 - 5)
Denominator: n^4 - 7n^2 + 10 = (n^2 - 2)(n^2 - 5)
Now simplify the expression:
(n^2 - 6)(n^2 - 5)/(n^2 - 2)(n^2 - 5)
= (n^2 - 6)/(n^2 - 2)
The simplified expression is n^2 - 6/n^2 - 2.
Since n^2 - 5 is a factor in both the numerator and denominator, n ≠ ± sqrt5.
The correct option is C: n^2 - 6/n^2 - 2; n ≠ ± sqrt5, n ≠ ± sqrt2.
Find the least common multiple of x³ – x² + x – 1 and x² – 1 . Write the answer in factored form.
(1 point)
Responses
a ( x + 1)²( x – 1)
b ( x + 1)( x – 1)( x² + 1)
c ( x³ – x² + x – 1)( x² – 1)
d (x + 1)(x – 1)(x² – 1)
To find the least common multiple (LCM) of x³ - x² + x - 1 and x² - 1, we need to factor both expressions first.
x³ - x² + x - 1 can be factored into (x - 1)(x² + 1).
x² - 1 can be factored further to (x - 1)(x + 1).
So, the factored form is:
(x - 1)(x² + 1) and (x - 1)(x + 1)
Now, the LCM is the product of all unique factors to their highest powers:
LCM = (x - 1)(x² + 1)(x + 1)
Therefore, the correct answer is b: (x + 1)(x - 1)(x² + 1).
Simplify the sum
7/a+8 + 7/a^2-64
To simplify the sum:
7/(a+8) + 7/(a²-64)
First, factor the denominator of the second fraction:
a² - 64 can be factored as (a+8)(a-8)
Now, we rewrite the expression:
7/(a+8) + 7/((a+8)(a-8))
To add these fractions, we need a common denominator. The common denominator is (a+8)(a-8).
So, we rewrite the fractions with the common denominator:
7(a-8)/((a+8)(a-8)) + 7/(a+8)(a-8)
Combine the numerators:
(7(a-8) + 7) / ((a+8)(a-8))
Simplify the numerator:
(7a - 56 + 7) / ((a+8)(a-8))
(7a - 49) / ((a+8)(a-8))
Therefore, the simplified sum is (7a - 49) / (a+8)(a-8).