1.Find the vertical asymptote of the rational function: f(x) = 3x-12 divided by 4x-2?

1.Find the vertical asymptote of the rational function: f(x) = 3x-12 divided by 4x-2


a. x = 1/2


b. x = 3/4


c. x = 2


d. x = 4





2. For the following equation, find the interval(s) where f(x) less than 0.


f(x) = 1 Divided by x^2-2x-8


a. (-4, 2)


b. (-2, 4)


c. (2, 4)


d. (2, 8)





3. Find the quotient and remainder of f(x) = x^3 - 4x^2 + 5x + 5 divided by p(x) = x - 1.


a. x^2 + 2x + 2; 7


b. x^2 - 3x + 3; -5


c. x^2 - 3x + 2; 7


d. x^2 - 2x + 3; -5





4. The polynomial f(x) divided x - 3 results in a quotient of x^2+3x-5 with a remainder of 2. Find f(3).


a) -5


b) -2


c) 2


d) 3





5. Let f(x) = x^3 - 8x^2 + 17x - 9. Use the factor theorem to find other solutions to f(x) - f(1) = 0, besides x = 1.


a) -2, 5


b) 2, -3


c) 2, 5


d) 2, 10

1.Find the vertical asymptote of the rational function: f(x) = 3x-12 divided by 4x-2?
1. Vertical asymptotes occur when the denominator is zero, i.e. when 4x-2 = 0, so at x = 1/2.





2. Factor the denominator x^2-2x-8 = (x-4)(x+2). Draw a number line and mark the points where the factors are zero, i.e. at x= 4 and -2. This divides the number line into three regions (a) x %26lt; -2 (b) -2 %26lt; x %26lt;4 and (c) x %26gt; 4.





When x %26lt; -2, (x-4) is negative and (x+2) is also negative so (x-4)(x+2) wil be positive, hence f(x) %26gt;0.





When -2 %26lt; x %26lt; 4, (x-4) is negative and (x+2) is positive so (x-4)(x+2) wil be negative, hence f(x) %26lt;0.





When x %26gt; 4, (x-4) is positive and (x+2) is also positive so (x-4)(x+2) wil be positive, hence f(x) %26gt;0.





3. What is x^3 divided by x? Answer is x^2 and multiplying p(x) by x^2 gives x^3-x^2. Subtract this from f(x) to get -3x^2+5x+5. Continue the division by mulitplying p(x) by -3x (to get rid of the -3x^2) to give -3x^2+3x and subtract from -3x^2+5x+5. This leaves 2x+5 and multiplying p(x) by 2 gives 2x-2; subtracting from 2x+5 leaves remainder 5-(-2)=7. Quotient is x^2-3x+2, remainder 7.





4. Straightforward application of remainder theorem, i.e. remainder after a polynomial f(x) is divided by (x-a) is f(a). So f(3) is 2.





5. Again using the factor (or remainder) theorem, we calculate f(2) = 2^3-8*2^2+17*2-9 = 8-32+34-9 = 1,


f(5) = 5^3-8*5^2+17*5-9 = 125-200+85-9 = 1 so answer is c.