SHORT ANSWER Write the word or phrase that best completes ea

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question Construct a rational expression with the given characteristics 3) The graph of R(x) crosses the x-axis at-1, touches the x-axis at 4, has vertical asymptotes 3) 2 at x =-2 and x = 3, and has one horizontal asymptote at y Find an nth degree polynomial function with real coefficients satisfying the given conditions. 4) n-4;2i, 4, and-4 are zeros; leading coefficient is1 4) a) Use the Factor Theorem to determine whether x -c is a factor of f b) If it is, write f in factored form, that is, write f in the form f(x) (x-c(quotient). 5) fx) x4+ 6x3 + 5x2 + 27x-18;c -6 5)

Solution

1) vertical asymptotes are at x = -2 and x = 3

zeros are x = -1 , -4

horizontal asymptote is y = -2

so , vertical asymptotes come in denominator and zeros in the numerator

we can write

R(x) = -2 ( x +1 ) ( x + 4)^2 / ( x+2)^2 ( x- 3)

simplifying we get

R(x) = ( - 2x^3 -18x^2-48x - 32 ) / ( x^3+x^2-8x-12 )