Pre-Calculus Review 1

This topic is a collection of problems and concepts that might help someone understand their working knowledge of Pre-Calculus 2.

Show all work and justify each step.

(1) Consider the polynomial function where

    (a) Sketch the graph.

    (b) What is the -intercept.

    (c) What is the solution to

    (d) What is the solution to

(2) Find the missing constants or zeros.

    (a) If find a number such that the graph of contains the point

    (b) If one zero of is find two other zeros.

    (b) If one zero of is find two other zeros.

(3) Sketch the graph of the function by (i) applying the Leading Coefficient Test, (ii) find the zeros of the polynomial, (iii) plotting sufficient solution points, and (iv) drawing a continuous curve through the points.

    (a)

    (b)

    (c)

    (d)

    (e)

    (f) [use part (e)]

(4) Use long division to divide

    (a)

    (b)

(5) Use synthetic division to divide

    (a)

    (b) pre calculus two review 1 _gr_23.gif]

(6) Express the function in the form for the given value of and demonstrate that

    (a) with

    (b) with

(7) Find a real number such that is a factor of

(8) Find the values for such that is divisible by the linear polynomial

(9) Show that is not a factor of for any real number

(10) Construct a cubic polynomial function with -intercepts of and which passes through the point

(11) Find the zeros of and state the multiplicity of each. Sketch the graph of each function

    (a)

    (b)

(12) Find a polynomial of degree 7 such that and are both zeros of multiplicity 2, 0 is a zero of multiplicity 3, and Sketch the graph of

(13) Construct a polynomial function with the stated properties.

    (a) Fifth degree, only real coefficients, 0 is the only real zero, is a zero of multiplicity 1, leading coefficient is 1.

    (b) Fourth degree, only real coefficients, -intercepts are 0 and 6, is a zero, leading coefficient is 3.

    (c) Fifth degree, is a zero of multiplicity 2, another integer is a zero of multiplicity 3, and -intercept is leading coefficient is 1.

(14) Find the remainder when

is divided by without using synthetic division or long division.

(15) List the roots of the polynomial

and state the multiplicity of each.

(16) Write as a product of linear factors.

(17) Determine a polynomial of lowest degree with roots 2 (of multiplicity 3), and with real coefficients.

(18) Determine a polynomial of lowest degree with roots (of multiplicity 2) and with real coefficients.

(19)  Determine a polynomial of lowest degree with roots and with integer coefficients.

(20) Solve the equation

(21) Determine the intercepts, asymptotes, and holes for the following rational functions. Also sketch and label the intersects and asymptotes on the graph. Point as many points as needed to obtain a rough sketch of the graph.

    (i)

    (ii)

    (iii)

    (iv)

    (v)

    (vi)

(22) Find the angle that is complementary to

(23) Express in terms of degrees, minutes, and seconds, to the nearest second.

(24) Express the angle as a decimal, to the nearest ten-thousandth of a degree.

(25) Show how to construct a unit circle step by step.

(26) Determine whether or for and Explain why.

(27) Find the radian measure of the smallest positive angle that is coterminal with

(28) Suppose is an angle of a right triangle, is the length of the side adjacent to and is the length of the hypotenuse. Find the values of the six trigonometric functions for the angle in terms of and

(29) Find the exact values of the trigonometric functions for the acute angle given

(30) Simplify and

(31) Simplify the expression

(32) Write the first expression in terms of the second, for any acute angle:

    (a)

    (b)

(33) For all values θ, verify

(34) Find the value of the six trigonometric functions given and

(35) The terminal side of angle passes through the intersection point of the given curves. Find the trigonometric functions of if they exist.

    (a) and

    (b) and