Exam 3: Polynomial and Rational Functions

Find a quadratic function whose graph is a parabola with $x$ - intercepts of $( - 2,0 )$ and $( 4,0 )$ and highest point $( h , 4 )$ .

Find the polynomial of degree $4$ that has zeros $- 1$ , $0$ , $1$ , and $5$ .

Find the polynomial of degree $5$ that has zeros $- 1$ , $2$ , and $3$ , with $- 1$ and $2$ being double roots.

Find all asymptotes (including vertical, horizontal, and slant) of the function $y = \frac { 3 x + 2 } { x ^ { 2 } - 2 x + 1 }$ .

Find all rational zeros of the polynomial $p ( x ) = x ^ { 4 } - 6 x ^ { 3 } + 54 x - 81$ .

Let $p ( x ) = x ^ { 4 } - 10 x ^ { 3 } + 37 x ^ { 2 } - 60 x + 36$ . Show that $p ( 2 ) = 0$ , and $p ( 3 ) = 0$ . Use this fact to find all the zeros of $p ( x )$ .

Factor the polynomial $p ( x ) = x ^ { 3 } - 8$ completely and find all its zeros. State the multiplicity of each zero.

Find all real zeros of the polynomial $p ( x ) = 4 x ^ { 3 } - 2 x - 4 x ^ { 2 } + 2$ . Use the quadratic formula if necessary.

Find intercepts and asymptotes, and then graph the rational function $y = \frac { 81 } { x ^ { 4 } - 16 }$ .

Use synthetic division and the remainder theorem to evaluate $p ( - 4 )$ for $p ( x ) = x ^ { 4 } - 256$ . 

Find all real zeros of the polynomial $p ( x ) = x ^ { 4 } - 6 x ^ { 3 } + 11 x ^ { 2 } - 8 x + 2$ . Use the quadratic formula if necessary.

Use a graphing device to find all real solutions of the equation. $x ^ { 5 } - 1.9 x ^ { 4 } - 0.85 x ^ { 3 } - 4.5 x ^ { 2 } + 9.00 x + 5.00 = 0$

Find the quotient and remainder of $\frac { 5 x ^ { 5 } + 6 x ^ { 3 } - 6 x ^ { 2 } - x ^ { 4 } + 2 x + 6 } { x ^ { 3 } + x - 1 }$ using long division.

Find a polynomial $P$ of degree $5$ , with integer coefficients that has zeros of $- \frac { 1 } { 2 }$ , $2$ , $i$ , with $2$ being a zero of multiplicity $2$ .

Find all solutions of the equation $6 x ^ { 2 } + 5 x = - 24$ and express them in standard form $a + b i$ .

Factor the polynomial $p ( x ) = x ^ { 6 } - 6 x ^ { 4 } + 9 x ^ { 2 }$ completely and find all its zeros. State the multiplicity of each zero.

Find all real and complex zeros of $P ( x ) = x ^ { 3 } - 2 x ^ { 2 } - 6 x - 8$

Use the Factor Theorem to show that $\left( x + \frac { 2 } { 3 } \right)$ is a factor of the polynomial $P ( x ) = x ^ { 3 } + \frac { 7 } { 6 } x ^ { 2 } - \frac { 1 } { 6 } x - \frac { 1 } { 3 }$ .

Find all rational zeros of the polynomial $p ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 11 x - 6$ .

Evaluate the expression and write the result in the form a + bi. $( 3 i ) ^ { 4 }$