Use factorization to simplify the given expression in part (a). Then find the indicated limit in part (b).
(a)
3x3 + 9x2 + 9x + 3
x4 + x3 + x + 1
(b)
lim
x → −1
3x3 + 9x2 + 9x + 1
x4 + x3 + x + 1
did you mean
3x^3 + 9x^2 + 9x + 3 ? If so, then
= 3(x+1)(x+1)(x+1)
x^4 + x^3 + x + 1
= x^3(x+1) + (x+1)
= (x+1)(x^3 + 1)
= (x+1)(x+1)(x^2 - x + 1)
so in your limit you would have
lim 3(x+1)^3 / ((x+1)^2(x^2 - x + 1) as x=-1
= lim 3(x+1)/(x^2 - x + 1)
= 0/3 = 0
(a) To simplify the given expression, we can try to factorize the numerator and denominator.
For the numerator, we have the polynomial 3x^3 + 9x^2 + 9x + 3. We can observe that each term in the polynomial has a factor of 3, so we can factor out 3:
3(x^3 + 3x^2 + 3x + 1)
Now, let's try to factorize the expression inside the parentheses. Unfortunately, this polynomial cannot be easily factorized further, as it does not seem to have any common factors or any recognizable factorization patterns.
Moving on to the denominator, we have the polynomial x^4 + x^3 + x + 1. Again, this polynomial does not seem to have any common factors or recognizable factorization patterns.
Therefore, the expression cannot be further simplified using factorization.
(b) To find the indicated limit, we substitute the given value (-1) into the expression and simplify.
Substituting x = -1 into the expression, we get:
lim (x → -1) [ 3x^3 + 9x^2 + 9x + 1 / x^4 + x^3 + x + 1]
= [3(-1)^3 + 9(-1)^2 + 9(-1) + 1] / [(-1)^4 + (-1)^3 + (-1) + 1]
= [-3 + 9 - 9 + 1] / [1 + (-1) + (-1) + 1]
= [-2] / [0]
Since the denominator is 0, the limit does not exist.