Simplify the rational expression. State any restrictions on the variable.
t^2 +2t -24/t^2-36
We can factor the numerator and denominator:
t^2 + 2t - 24 = (t+6)(t-4)
t^2 - 36 = (t+6)(t-6)
So the expression simplifies to:
(t+6)(t-4)/[(t+6)(t-6)]
Simplifying further, we can cancel out the (t+6) terms to get:
(t-4)/(t-6)
Therefore, the simplified expression is (t-4)/(t-6). However, we need to note that the original expression is undefined when t = 6 or t = -6, since this would make the denominator equal to zero. So the restriction on the variable is that t cannot equal 6 or -6.
What is the product in simplest form? State any restrictions on the variable.
x^2 + 7x+10/x+3 times x^2-3x-18/x^2+x-2
A (x+5)(x-6)/x-1, x ≠ 3, x ≠ -2, x ≠ 1
B (x+5)(x-6)/x-1, x ≠-2, x ≠ 1
We first need to factor each expression:
x^2 + 7x + 10 = (x + 5)(x + 2)
x + 3 can't be factored further
x^2 - 3x - 18 = (x - 6)(x + 3)
x^2 + x - 2 = (x - 1)(x + 2)
Plugging these into the expression and simplifying, we get:
[(x + 5)(x + 2)/(x + 3)] * [(x - 6)(x + 3)/(x - 1)(x + 2)]
We can then cancel out the (x + 2) and (x + 3) terms:
[(x + 5)/(x - 1)] * [(x - 6)/(x + 2)]
The final simplified form of the expression in simplest form is:
(x + 5)(x - 6)/(x - 1)(x + 2)
The restrictions on the variable are that x cannot be equal to -2, 1, or 3 since these would make the denominator equal to zero. Therefore, the correct answer is B, with the specified restrictions.
Simplify the sum --> d^2-9d+20/d^2-3d -10 + d^2-2d-8/d^2+4d-32
A. 2D^2 +8D-28/(D+2)(D+8)
First, let's factor the expressions in each fraction:
d^2 - 9d + 20 = (d - 5)(d - 4)
d^2 - 3d - 10 = (d - 5)(d + 2)
d^2 - 2d - 8 = (d - 4)(d + 2)
d^2 + 4d - 32 = (d + 8)(d - 4)
Now, we can write the sum with the common denominators:
[(d - 5)(d - 4)/(d - 5)(d + 2)] + [(d - 4)(d + 2)/(d + 8)(d - 4)]
We can simplify this by canceling out the common factors:
[d - 4)/(d + 2)] + [(d + 2)/(d + 8)]
Now we can add these two fractions by finding a common denominator:
[(d - 4)(d + 8) + (d + 2)(d + 2)] / [(d + 2)(d + 8)]
Expanding and collecting like terms:
(2d^2 + 8d - 28) / [(d + 2)(d + 8)]
The final answer, with the expression simplified, is:
2d^2 + 8d - 28 / (d + 2)(d + 8)
There are no restrictions on the variable. Therefore, the correct answer is A.
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
Does the series converge or diverge? If it converges, what is the sum? Show your work. ∑∞ n=1 -4(-1/2)^n-1
We can start by looking at the general formula for a geometric series:
∑∞ n=1 ar^(n-1)
Where a is the first term and r is the common ratio.
In this case, we have:
a = -4
r = -1/2
So, our series becomes:
∑∞ n=1 -4(-1/2)^(n-1)
We can simplify this expression by factoring out the constant -4, and then using the formula for the sum of an infinite geometric series:
∑∞ n=1 -4(-1/2)^(n-1) = -4 * ∑∞ n=1 (-1/2)^(n-1)
The formula for the sum of an infinite geometric series is:
sum = a / (1 - r)
Where a is the first term and r is the common ratio.
In our case, a = 1 and r = -1/2. So:
sum = 1 / (1 - (-1/2)) = 1 / (3/2) = 2/3
Thus, the sum of the given series converges to 2/3.