Question 1

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

Accepted Answer

y-intercept: $\left( 0 , - \frac { 1 } { 3 } \right)$

Question 2

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

Accepted Answer

$( x + 5 i ) ( x - 5 i ) ( x + \sqrt { 5 } ) ( x - \sqrt { 5 } )$

Question 3

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

Accepted Answer

$x = 3 \pm i , 2 - i , 0$

Question 4

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.
​

Accepted Answer

451.98 vib

Question 5

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

Accepted Answer

The answer of Find all the zeros of the function...

Question 6

The cost C (in millions of dollars) of removing p% of the industrial and municipal pollutants discharged into a river is given by
​ $C = \frac { 250 p } { 100 - p } , 0 \leq p \leq 100$ .
​
Select the correct graph of the cost function.
​

Accepted Answer

The answer of The cost C (in millions of dollars)...

Question 7

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

Accepted Answer

Domain of f: all rea

Question 8

Find the zeros (if any) of the rational function.
​ $g ( x ) = \frac { x ^ { 2 } - 4 } { x + 2 }$ ​

Accepted Answer

The answer of Find the zeros (if any) of the...

Question 9

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.

Accepted Answer

The answer of Given that \(x = 3 - 5...

Question 10

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

Accepted Answer

The answer of Find all the zeros of the function.
​...

Question 11

Find a mathematical model representing the statement. (Determine the constant of proportionality.)
​
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 $s = 1.8$ .)
​

Accepted Answer

The answer of Find a mathematical model representing the statement....

Question 12

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 $x - 3$ .
​

Accepted Answer

The answer of Use synthetic division to express \(P (...

Question 13

Use long division to divide.
​ $\left( x ^ { 3 } - 216 \right) \div ( x - 6 )$ ​

Accepted Answer

The answer of Use long division to divide.
​ \(\left( x...

Question 14

Find a polynomial function with real coefficients that has the given zeros.
​
2, $4 + i$ ​

Accepted Answer

The answer of Find a polynomial function with real coefficients...

Question 15

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

Accepted Answer

The answer of Use synthetic division to divide. \(\left( -...

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

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

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

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

The cost C (in millions of dollars) of removing p% of the industrial and municipal pollutants discharged into a river is given by $C = \frac { 250 p } { 100 - p } , 0 \leq p \leq 100$ . Select the correct graph of the cost function.

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

Find the zeros (if any) of the rational function. $g ( x ) = \frac { x ^ { 2 } - 4 } { x + 2 }$

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

Find a mathematical model representing the statement. (Determine the constant of proportionality.) 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 $s = 1.8$ .)

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 $x - 3$ .

Use long division to divide. $\left( x ^ { 3 } - 216 \right) \div ( x - 6 )$

Find a polynomial function with real coefficients that has the given zeros. 2, $4 + i$

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

Topics From Algebra

Functions and Graphs

Inverse, Exponential, and Logarithmic Functions

The Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems of Equations and Inequalities

Sequences, Series, and Probability

Topics From Analytic Geometry

Limits of Functions

Filters

Exam 3: Polynomial and Rational Functions

Find all intercepts of the following function. ​ g(x)=1x−3g ( x ) = \frac { 1 } { x - 3 }g(x)=x−31​ ​

Write the polynomial in completely factored form. ​ f(x)=x4+20x2−125f ( x ) = x ^ { 4 } + 20 x ^ { 2 } - 125f(x)=x4+20x2−125 ​

Given 2+i2 + i2+i is a root, determine all other roots of f(x)=x5−10x4+39x3−70x2+50xf ( x ) = x ^ { 5 } - 10 x ^ { 4 } + 39 x ^ { 3 } - 70 x ^ { 2 } + 50 xf(x)=x5−10x4+39x3−70x2+50x .

Find all the zeros of the function and write the polynomial as a product of linear factors. ​ x4+65x2+784x ^ { 4 } + 65 x ^ { 2 } + 784x4+65x2+784 ​

The cost C (in millions of dollars) of removing p% of the industrial and municipal pollutants discharged into a river is given by ​ C=250p100−p,0≤p≤100C = \frac { 250 p } { 100 - p } , 0 \leq p \leq 100C=100−p250p​,0≤p≤100 . ​ Select the correct graph of the cost function. ​

Determine the domains of f and g. ​ f(x)=7x−17x2−x,g(x)=1xf ( x ) = \frac { 7 x - 1 } { 7 x ^ { 2 } - x } , g ( x ) = \frac { 1 } { x }f(x)=7x2−x7x−1​,g(x)=x1​ ​

Find the zeros (if any) of the rational function. ​ g(x)=x2−4x+2g ( x ) = \frac { x ^ { 2 } - 4 } { x + 2 }g(x)=x+2x2−4​ ​

Given that x=3−5ix = 3 - 5 ix=3−5i is a zero of f(x)=x3−10x2+58x−136f ( x ) = x ^ { 3 } - 10 x ^ { 2 } + 58 x - 136f(x)=x3−10x2+58x−136 , find all the zeros of f.

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

Find a mathematical model representing the statement. (Determine the constant of proportionality.) ​ v varies jointly as p and q and inversely as the square of s. ( v=1.7v = 1.7v=1.7 when p=5.7p = 5.7p=5.7 , q=7.2q = 7.2q=7.2 and s=1.8s = 1.8s=1.8 .) ​

Use synthetic division to express P(x)=5x3−8x2−19x−23P ( x ) = 5 x ^ { 3 } - 8 x ^ { 2 } - 19 x - 23P(x)=5x3−8x2−19x−23 in the form (divisor)(quotient) + remainder for the divisor x−3x - 3x−3 . ​

Use long division to divide. ​ (x3−216)÷(x−6)\left( x ^ { 3 } - 216 \right) \div ( x - 6 )(x3−216)÷(x−6) ​

Find a polynomial function with real coefficients that has the given zeros. ​ 2, 4+i4 + i4+i ​

Use synthetic division to divide. (−8+3x3−2x+7x2)÷(x+2)\left( - 8 + 3 x ^ { 3 } - 2 x + 7 x ^ { 2 } \right) \div ( x + 2 )(−8+3x3−2x+7x2)÷(x+2)

Topics From Algebra

Functions and Graphs

Inverse, Exponential, and Logarithmic Functions

The Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems of Equations and Inequalities

Sequences, Series, and Probability

Topics From Analytic Geometry

Limits of Functions

Filters

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

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

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

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

The cost C (in millions of dollars) of removing p% of the industrial and municipal pollutants discharged into a river is given by $C = \frac { 250 p } { 100 - p } , 0 \leq p \leq 100$ . Select the correct graph of the cost function.

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

Find the zeros (if any) of the rational function. $g ( x ) = \frac { x ^ { 2 } - 4 } { x + 2 }$

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

Find a mathematical model representing the statement. (Determine the constant of proportionality.) 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 $s = 1.8$ .)

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 $x - 3$ .

Use long division to divide. $\left( x ^ { 3 } - 216 \right) \div ( x - 6 )$

Find a polynomial function with real coefficients that has the given zeros. 2, $4 + i$

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