Sketch a graph of y fx For each of the following polynomial

Sketch a graph of y = f(x). For each of the following polynomials p(x): p(x) = x^2 - 3x^2 + 4 p(x) = -x^3 + 4x^2 - x - 6 p(x) = 2x^4 + 7x^3 + 6 x^2 - x - 2 List all possible rational roots of p(x), according to the Rational Zeros Theorem. Factor p(x) completely. Find all roots of the equation p(x) = 0. Determine the end behaviour of the graph of y = p(x). Determine the y - intercept of the graph of y = p(x). Determine the x - intercepts of the graphs y = p(x) Determine the local behaviour of y = p(x) near the x - intercepts. Use the above information to sketch a graph of y = p(x). Find the remainder of the division of x^122 - 20 x^31 + 60 x^34 + 1 when divided by x - 1. For each of the following rational functions f. f(x) = x^2 + 2x + 1/x^2 - x - 2 f(x) = x^2 + 2x - 3/x^2 - 2x -3 f(x) = x^2 - 9/x^2 -x -2 f(x) = 2 - x/x^2 + x -2 f(x) = x^2/x^2 +1 Factor numerator and denominator and simplify if possible. Find the x and y intercepts of the graph of y = f(x) if they exists. Find any vertical or horizontal asymptotes. Determine how the sign of f(x) changes. Use the above information to sketch a graph of y = f(x). Solve the following inequalities. Express your answer using interval notation. x^4 + x^3 - 7x^2 - x + 6 Greaterthanorequalto 0 x + 4/2x - 1 > 3 x^2 - 3x + 2/x^3 - 6x^2 + 9x

Solution

12) a) log(1/64) = log(1/2) = log2 = -6

b) log(1/3) = (-1/2)*log 3

c) log_b x^3y = 3*log_b x + log_b y = 3*2 + 36 = 42

d) e^x-y = e^x *e^-y = 3/4

e) log_a(x/y) = log_a x/log_a y = 12/4 = 3

f) ln e² = 2*lne = 2

g) log 1000 = log 10³ = 3

h) log₇31 = log 31/log7 = 1.7647

i) sin^-1(sin pi/6) = pi/6

j) cos^-1(cos 4pi/3) = 4pi/3

k) cos(sin^-1 (-1)) = cos(-90) = 0