Deck 3: Polynomial and Rational Functions

The cost C (in millions of dollars) of removing p% of the industrial and municipal pollutants discharged into a river is given by
​ $C = \frac { 250 p } { 100 - p } , 0 \leq p \leq 100$ .
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Select the correct graph of the cost function.
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The frequency of vibrations of a piano string varies directly as the square root of the tension on the string and inversely as the length of the string. The middle A string has a frequency of 410 vibrations per second. Find the frequency of a string that has 1.75 times as much tension and is 1.2 times as long.
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Given that $x = 3 - 5 i$ is a zero of $f ( x ) = x ^ { 3 } - 10 x ^ { 2 } + 58 x - 136$ , find all the zeros of f.

Find all intercepts of the following function.
​ $g ( x ) = \frac { 1 } { x - 3 }$ ​

Find all the zeros of the function.
​ $( x - 7 ) ( x + 8 ) ^ { 3 }$ ​

Find a mathematical model representing the statement. (Determine the constant of proportionality.)
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v varies jointly as p and q and inversely as the square of s. ( $v = 1.7$ when $p = 5.7$ , $q = 7.2$ and $s = 1.8$ .)
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Use long division to divide.
​ $\left( x ^ { 3 } - 216 \right) \div ( x - 6 )$ ​

Use synthetic division to divide. $\left( - 8 + 3 x ^ { 3 } - 2 x + 7 x ^ { 2 } \right) \div ( x + 2 )$

Find all the zeros of the function and write the polynomial as a product of linear factors.
​ $x ^ { 4 } + 65 x ^ { 2 } + 784$ ​

Determine the domains of f and g.
​ $f ( x ) = \frac { 7 x - 1 } { 7 x ^ { 2 } - x } , g ( x ) = \frac { 1 } { x }$ ​

Use synthetic division to express $P ( x ) = 5 x ^ { 3 } - 8 x ^ { 2 } - 19 x - 23$ in the form (divisor)(quotient) + remainder for the divisor $x - 3$ .
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Find a polynomial function with real coefficients that has the given zeros.
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2, $4 + i$ ​

Given $2 + i$ is a root, determine all other roots of $f ( x ) = x ^ { 5 } - 10 x ^ { 4 } + 39 x ^ { 3 } - 70 x ^ { 2 } + 50 x$ .

Write the polynomial in completely factored form.
​ $f ( x ) = x ^ { 4 } + 20 x ^ { 2 } - 125$ ​