Exam 2: Polynomial and Rational Functions

Find a polynomial function with following characteristics. Degree: 4 Zero: 4 , multiplicity: 2 Zero: $- 3$ , multiplicity: 2 Falls to the left, Falls to the right Absolute value of the leading coefficient is one

Determine the $x$ -intercept(s) of the quadratic function $f ( x ) = x ^ { 2 } - 15 x + 56$ .

Find a fifth degree polynomial function of the lowest degree that has the zeros below and whose leading coefficient is one. $- 3 , - 1,0,1,3$

Use synthetic division to divide. $\left( 6 + 5 x ^ { 3 } + 23 x + 22 x ^ { 2 } \right) \div ( x + 3 )$

Match the equation with its graph. $f ( x ) = x ^ { 2 } + 2 x - 3$

Find all real zeros of the polynomial $f ( x ) = x ^ { 4 } + 13 x ^ { 3 } + 40 x ^ { 2 }$ and determine the mutiplicity of each.

Determine the $x$ -intercept(s) of the quadratic function $f ( x ) = x ^ { 2 } + 6 x + 10$ .

Using the factors $( x + 4 )$ and $( x + 3 )$ , find the remaining factor(s) of $f ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 5 x - 12$ and write the polynomial in fully factored form.

If $f ( x ) = 4 x ^ { 2 } - 2 x - 7$ , use synthetic division to evaluate $f \left( \frac { 7 } { 8 } \right)$ .

Write the complex conjugate of the complex number $- 5 - \sqrt { 10 } i$ .

The interest rates that banks charge to borrow money fluctuate with the economy. The interest rate charged by a bank in a certain country is given in the table below. Let $t$ represent the year, with $t = 0$ corresponding to 1986 . Use the regression feature of a graphing utility to find a quadratic model of the form $y = a t ^ { 2 } + b t + c$ for the data. Year Percent 1986 12.4 1988 9.7 1990 7.3 1992 6.3 1994 9.7 1996 11.6

Find real numbers $a$ and $b$ such that the equation $a + b i = - 10 + 10 i$ is true.

Find the zeros of the function below algebraically, if any exist. $f ( x ) = 25 x ^ { 3 } - 60 x ^ { 2 } + 36 x$

Determine the zeros (if any) of the rational function $g ( x ) = 7 + \frac { 2 } { x ^ { 2 } + 7 }$ .

Simplify $\frac { 1 - 2 i } { 5 i }$ and write the answer in standard form.

Find a polynomial function of the lowest degree with real coefficients that has the zeros below and whose leading coefficient is one. $0 , - 3,1 + 3 i$

Suppose the IQ scores $( y$ , rounded to the nearest $10 )$ for a group of people are summarized in the table below. Use the regression feature of a graphing utility to find a quadratic function of the form $y = a x ^ { 2 } + b x + c$ for the data. IQ Score Number of People 70 53 80 72 90 93 100 90 110 78 120 47 130 16

Write the complex conjugate of the following complex number and then multiply the number by the complex conjugate. Write the result in standard form. $1 + \sqrt { - 20 }$

Find all real zeros of the polynomial $f ( x ) = x ^ { 4 } - 80 x ^ { 2 } + 1024$ and determine the mutiplicity of each.

Simplify $( \sqrt { - 2 } ) ^ { 11 }$ and write the answer in standard form.