I need help Math friends ?

Question

Consider a rational function whose denominator is -x2 + 4x + ½. What is the maximum number of vertical asymptotes it may have?

a)2

b)4

c)1/2

d) The denominator defines the function’s zeroes, not its vertical asymptotes

When factoring the following by the reverse FOIL method, the factors of 6 must add up to the value of B in f(x) = x2 + Bx + 6; what are the possible values of B?

a) 5,7

b)+5, -5, +7, -7

c) -5, 7

D) None of the above

Answer 1

Only zeroes of the denominator can correspond to vertical asymptotes. Since

-x^2 + 4x + 1/2 = - (x - 2)^2 + 9/2 = 0

has two distinct solutions (namely x = 2 + 3/sqrt(2) and x = 2 - 3/sqrt(2)), it follows that the maximum number of vertical asymptotes is 2

Answer a)

For the FOIL method, call y and z the factors of 6. Then you have

x^2 + Bx + 6 = (x+y)(x+z) with

yz = 6 and

y + z = B

Note that y = 1 and z = 6 yields

yz = 6 and y + z = 7 = B

Similarly, note that y = - 1 and z = - 6 yields

yz = 6 and y + z = - 7 = B

Now, note that y = 2 and z = 3 yields

yz = 6 and y + z = 5 = B

Similarly, note that y = - 2 and z = - 3 yields

yz = 6 and y + z = - 5 = B

Hence B can be 5, -5, 7 or -7.

The answer is b)