Question 1

Write the complex number in standard form. $\sqrt { - 9 }$
A) $3 i$
B) $- 3 i$
C) $9 i$
D) $4 i$
E) $- 9 i$

Accepted Answer

The square root of a negative number involves the imaginary unit $i$, where $i = \sqrt{-1}$. Thus, $\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i$.

Question 2

Select the graph of the function $y = - 3 x ^ { 2 }$ . Compare the graph of this function with the graph of $y = x ^ { 2 }$ 
A)

Vertical stretch

B)

Vertical stretch and reflection in $x$-axis

C)

Reflection in $X$ -axis

D)

Vertical shrink and reflection in $x$-axis

E)

Vertical shrink

Accepted Answer

The graph of $y=-3x^2$ is a vertical stretch of the graph of $y=x^2$ by a factor of 3 and a reflection in the x-axis. This is because the negative sign in front of the 3 causes the graph to be reflected in the x-axis, and the absolute value of the coefficient 3 gives the factor of vertical stretch/shrink.

Question 3

Select the correct graph of the function given by $f ( x ) = x ^ { 4 }$ . Explain how the graph of function g differs (if it does) from the graph of each function $f$. $g ( x ) = f ( x ) + 8$ 
A)

Horizontal shift of eight units

B)

Vertically downward shift of eight units
C)

Horizontal shift of eight units to the 
D)

Vertical shift of eight units

E)

Horizontal shift of eight units to the left right

Accepted Answer

D

Question 4

Use synthetic division to divide. 
$\frac { x ^ { 3 } - 216 } { x - 6 }$
A) $x ^ { 2 } + 6 x + 36 , x \neq 6$
B) $x ^ { 2 } + 6 x - 36 , x \neq - 6$
C) $x ^ { 2 } - 6 x + 36 , x \neq - 6$
D) $- x ^ { 2 } + 6 x + 36 , x \neq - 6$
E) $x ^ { 2 } - 6 x - 36 , x \neq - 6$

Accepted Answer

Using synthetic division with $x - 6$, the coefficients of $x^3 - 216$ become $1, 0, 0, -216$. Dividing by $x = 6$, we get $1(6)^2 + 6(6) + 36 = x^2 + 6x + 36$, confirming choice A is correct.

Question 5

Solve the inequality and graph the solution on the real number line $x ^ { 2 } \leq 16$
A) $( - 4,4 )$

B) $[ - 4,4 ]$

C) $[ - 4,4 ]$

D) $[ 4 , - 4 ]$

E) $( - 4,4 )$

Accepted Answer

To solve the inequality we can take the square root of both sides: $x \leq \pm4$, which means $x$ is less than or equal to 4 and greater than or equal to $-4$. Therefore, the solution is the interval $[-4,4]$ which is represented on the number line by choice B.

Question 6

Write the quadratic function $f ( x ) = - x ^ { 2 } + 2 x + 2$ in standard form. 
A) $f ( x ) = ( x - 3 ) ^ { 2 } - 1$
B) $f ( x ) = ( x - 1 ) ^ { 2 } - 3$
C) $f ( x ) = - ( x + 1 ) ^ { 2 } + 3$
D) $f ( x ) = - ( x + 3 ) ^ { 2 } - 1$
E) $f ( x ) = - ( x - 1 ) ^ { 2 } + 3$

Accepted Answer

To write a quadratic function in standard form, we need to expand the squared term, combine the constant terms, and put the terms in descending order of degree.

Starting with $f(x)=-x^2+2x+2$, we can complete the square as follows:

\begin{align*}
f(x)&=-x^2+2x+2\
&=-(x^2-2x)+2\
&=-(x^2-2x+1-1)+2\
&=-[(x-1)^2-1]+2\
&=-(x-1)^2+3
\end{align*}

Putting the function in standard form, we get:

$$f(x)=-1(x-1)^2+3$$

which matches choice (E).

Question 7

Perform the operation and write the result in standard form. 
$\frac { 4 } { 1 + i } - \frac { 5 } { 1 - i }$
A) $- \frac { 1 } { 2 } + \frac { 2 } { 9 } i$
B) $- \frac { 1 } { 2 } - \frac { 2 } { 9 } i$
C) $- \frac { 1 } { 2 } + \frac { 9 } { 2 } i$
D) $\frac { 9 } { 2 } - \frac { 1 } { 2 } i$
E) $- \frac { 1 } { 2 } - \frac { 9 } { 2 } i$

Accepted Answer

The answer of Perform the operation and write the result...

Question 8

Select the correct graph of the following function. 
$f ( x ) = \frac { x ^ { 2 } - 49 } { x + 7 }$
A)

B)

C)

D)

E)

Accepted Answer

The answer of Select the correct graph of the following...

Question 9

If $x = \sqrt { 7 } \text { is a root of } x ^ { 3 } - 3 x ^ { 2 } - 7 x + 21 = 0$ , use synthetic division to factor the polynomial completely and list all real solutions of the equation. 
A) $( x - 3 ) ( x - \sqrt { 7 } ) ( x + \sqrt { 7 } ) ; 3 , \sqrt { 7 } , - \sqrt { 7 }$
B) $( x + 3 ) ( x + \sqrt { 7 } ) ^ { 2 } ; - 3 , \sqrt { 7 }$
C) $( x + 3 ) ( x - \sqrt { 7 } ) ^ { 2 } ; - 3 , \sqrt { 7 }$
D)$( x + 3 ) ( x + \sqrt { 7 } ) ^ { 2 } ; 3 , - \sqrt { 7 }$
E) $( x + 3 ) ^ { 2 } ( x - \sqrt { 7 } ) ; - 3 , \sqrt { 7 }$

Accepted Answer

Given $x = \sqrt{7}$ is a root, synthetic division can be used to factor the polynomial. The factored form $(x - 3)(x - \sqrt{7})(x + \sqrt{7})$ indicates the roots are $3, \sqrt{7}, -\sqrt{7}$, which matches option A.

Question 10

Find all the zeros of the function and write the polynomial as a product of linear fac- tors. 
$y ^ { 4 } - 2401$ 
A) $\pm 7 ( y - 7 ) ( y + 7 )$
B) $\pm 7 , \pm 7 i ; ( y - 7 ) ( y + 7 ) ( y - 7 i ) ( y + 7 i )$
C) $7,7 i ; ( y + 7 ) ( y + 7 ) ( y - 7 i ) ( y - 7 i )$
D) $\pm 7 i ; ( y - 7 i ) ( y + 7 i )$
E) $\pm 7 , \pm 7 i ; ( y - 7 ) ( y - 7 ) ( y + 7 i ) ( y + 7 i )$

Accepted Answer

To find the zeros of the function, we can set it equal to zero and solve for y:

$$y^4-2401=0$$
$$(y^2)^2-(49)^2=0$$
$$(y^2-49)(y^2+49)=0$$

So the zeros are $y= \pm 7$ and $y= \pm 7i$.

To write the polynomial as a product of linear factors, we can use these zeros. The factors corresponding to the real zeros are simply $(y-7)$ and $(y+7)$. The factors corresponding to the complex zeros are $(y-7i)$ and $(y+7i)$.

Therefore, the polynomial can be expressed as:

$$(y^2-49)(y^2+49)=(y-7)(y+7)(y-7i)(y+7i)$$

So the best choice is B, which correctly identifies the zeros and the corresponding linear factors.

Question 11

The total revenue $R$ earned per day (in dollars) from a pet-sitting service is given by 
$R ( p ) = - 12 p ^ { 2 } + 150 p$ where, $p$ is the price charged per pet (in dollars). Find the revenue when the price per pet is $\$ 4$.
A) $\$ 408$
B) $\$ 438$
C) $\$ 448$
D) $\$ 428$
E) $\$ 418$

Accepted Answer

To find the revenue when the price per pet is $4, substitute $p = 4$ into the revenue equation: $R(4) = -12(4)^2 + 150(4) = -12(16) + 600 = -192 + 600 = \$408$.

Question 12

Simplify the complex number and write it in standard form. $\frac { 1 } { ( 6 i ) ^ { 3 } }$ 
A) $\frac { 1 } { 6 } i$
B)$- \frac { 1 } { 216 } i$
C) $\frac { 1 } { 216 } i$
D) $- \frac { 1 } { 6 } i$
E) $216 i$

Accepted Answer

The answer of Simplify the complex number and write it...

Question 13

Identify all intercepts of $f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 36 }$ 
A) $x$-intercept: none; $y$-intercept: $( 0,7 )$
B) $x$-intercept: none; $y$-intercept: $( 0,1 )$
C) $x$-intercept: none; $y$-intercept: none
D) $x$-intercept: $( 0,0 ) ; y$-intercept: $( 0,0 )$
E) $x$-intercepts: $( - 6,0 )$ and $( 6,0 ) ; y$-intercept: $( 0,1 )$

Accepted Answer

The answer of Identify all intercepts of \(f ( x...

Question 14

The cost C (in millions of dollars) removing p% the industrial and municipal pollu- tants discharged into a river is given by $C = \frac { 245 p } { 100 - p } , 0 \leq p \leq 100$ Select the correct graph of the cost function. 
A)

B)

C)

D)

E)

Accepted Answer

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

Question 15

Use long division to divide. 
$\left( x ^ { 4 } + 9 x ^ { 3 } + 20 x ^ { 2 } - x - 5 \right) \div ( x + 5 )$ 
A) $x ^ { 3 } + 4 x ^ { 2 } + 1$
B) $x ^ { 3 } - 4 x ^ { 2 } - 1$
C) $x ^ { 3 } + 4 x ^ { 2 } - 1$
D) $x ^ { 3 } + 4 x ^ { 2 } + 1$.
E) $x ^ { 3 } - 4 x ^ { 2 } + 1$.

Accepted Answer

We can use long division as shown below:

x^3 + 4x^2 - x
x+5 ) x^4 + 9x^3 + 20x^2 - x - 5
        x^4 + 5x^3           
            ----------------       
                 4x^3 + 20x^2    
                 4x^3 + 20x^2           
                          --------------             
                                  - x - 5               
                                  - x - 5               
                                      -------                   
                                            0

Therefore, the quotient is $x^3 + 4x^2 - x - 5/(x+5) = \boxed{x^3 + 4x^2 - 1}$. Option C is the only one that matches this answer.

Question 16

Find all the rational zeros of the function. 
$x ^ { 3 } - 15 x ^ { 2 } + 75 x - 125$
A) $- 5$
B) $0 , - 1$
C) 0,5
D) 5
E) $\pm 5$

Accepted Answer

The rational zeros of a polynomial can be found using the Rational Root Theorem, which suggests that any rational root, in the form of a fraction p/q, is a divisor of the constant term (in this case, -125) divided by a divisor of the leading coefficient (in this case, 1). For this polynomial, the possible rational roots are divisors of -125, which are ±1, ±5, ±25, ±125. Testing these values or using synthetic division, we find that 5 is a root because substituting x = 5 into the polynomial gives 0. None of the other options are roots of the polynomial.

Question 17

Use long division to divide. 
$\left( x ^ { 3 } + 5 x ^ { 2 } + 16 x + 80 \right) \div ( x + 5 )$
A) $x ^ { 2 } + 16$
B) $x ^ { 2 } + 10 x + 46 - \frac { 246 } { x + 5 }$
C) $x ^ { 2 } + 20$
D) $x ^ { 2 } + 10 x + 66 + \frac { 90 } { x + 5 }$
E) $x ^ { 2 } + 10 x + 46$

Accepted Answer

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

Question 18

Perform the operation and write the result in standard form. $( 10 - 9 i ) ^ { 2 }$
A) $19 - 180 i$
B) $- 19 - 180 i$
C) $19 + 180 i$
D) $21 - 180 i$
E) $- 19 + 180 i$

Accepted Answer

Expanding $(10 - 9i)^2$ gives $10^2 - 2 \cdot 10 \cdot 9i + (9i)^2 = 100 - 180i - 81 = 19 - 180i$.

Question 19

Find all the rational zeros of the function 
$f ( x ) = 2 x ^ { 5 } + 3 x ^ { 4 } - 9 x ^ { 3 } - 3 x ^ { 2 } - 11 x - 6$ 
A)$x = - \frac { 1 } { 2 } , - 3,2$
B) $x = \frac { 2 } { 3 } , - 1,2$
C) $x = \frac { 1 } { 2 } , - 3,1$
D) $x = - \frac { 1 } { 2 } , - \frac { 3 } { 2 }$
E) $x = - \frac { 1 } { 2 } , - \frac { 3 } { 2 } , 2$

Accepted Answer

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

Question 20

Solve the inequality and write the solution set in interval notation. 
$x ^ { 4 } ( x - 3 ) \leq 0$ 
A) $( - \infty , 3 ]$
B) $( \infty , 3 )$
C) $( - 3,3 ]$
D) $( \infty , 3 ]$
E) $( - \infty , - 3 ]$

Accepted Answer

The inequality $x^4(x - 3) \leq 0$ is satisfied when $x - 3 \leq 0$, because $x^4 \geq 0$ for all real x (since any real number raised to the fourth power is non-negative). Thus, the solution is $x \leq 3$, which in interval notation is $(- \infty , 3 ]$.

Quiz 2: Polynomial and Rational Functions

Write the complex number in standard form. $\sqrt { - 9 }$

Select the graph of the function $y = - 3 x ^ { 2 }$ . Compare the graph of this function with the graph of $y = x ^ { 2 }$

Select the correct graph of the function given by $f ( x ) = x ^ { 4 }$ . Explain how the graph of function g differs (if it does) from the graph of each function $f$ . $g ( x ) = f ( x ) + 8$

Use synthetic division to divide. $\frac { x ^ { 3 } - 216 } { x - 6 }$

Solve the inequality and graph the solution on the real number line $x ^ { 2 } \leq 16$

Write the quadratic function $f ( x ) = - x ^ { 2 } + 2 x + 2$ in standard form.

Perform the operation and write the result in standard form. $\frac { 4 } { 1 + i } - \frac { 5 } { 1 - i }$

Select the correct graph of the following function. $f ( x ) = \frac { x ^ { 2 } - 49 } { x + 7 }$

If $x = \sqrt { 7 } \text { is a root of } x ^ { 3 } - 3 x ^ { 2 } - 7 x + 21 = 0$ , use synthetic division to factor the polynomial completely and list all real solutions of the equation.

Find all the zeros of the function and write the polynomial as a product of linear fac- tors. $y ^ { 4 } - 2401$

The total revenue $R$ earned per day (in dollars) from a pet-sitting service is given by $R ( p ) = - 12 p ^ { 2 } + 150 p$ where, $p$ is the price charged per pet (in dollars) . Find the revenue when the price per pet is $\$ 4$ .

Simplify the complex number and write it in standard form. $\frac { 1 } { ( 6 i ) ^ { 3 } }$

Identify all intercepts of $f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 36 }$

The cost C (in millions of dollars) removing p% the industrial and municipal pollu- tants discharged into a river is given by $C = \frac { 245 p } { 100 - p } , 0 \leq p \leq 100$ Select the correct graph of the cost function.

Use long division to divide. $\left( x ^ { 4 } + 9 x ^ { 3 } + 20 x ^ { 2 } - x - 5 \right) \div ( x + 5 )$

Find all the rational zeros of the function. $x ^ { 3 } - 15 x ^ { 2 } + 75 x - 125$

Use long division to divide. $\left( x ^ { 3 } + 5 x ^ { 2 } + 16 x + 80 \right) \div ( x + 5 )$

Perform the operation and write the result in standard form. $( 10 - 9 i ) ^ { 2 }$

Find all the rational zeros of the function $f ( x ) = 2 x ^ { 5 } + 3 x ^ { 4 } - 9 x ^ { 3 } - 3 x ^ { 2 } - 11 x - 6$

Solve the inequality and write the solution set in interval notation. $x ^ { 4 } ( x - 3 ) \leq 0$

Quiz 2: Polynomial and Rational Functions

Write the complex number in standard form. −9\sqrt { - 9 }−9​

Select the graph of the function y=−3x2y = - 3 x ^ { 2 }y=−3x2 . Compare the graph of this function with the graph of y=x2y = x ^ { 2 }y=x2

Select the correct graph of the function given by f(x) =x4f ( x ) = x ^ { 4 }f(x) =x4 . Explain how the graph of function g differs (if it does) from the graph of each function fff . g(x) =f(x) +8g ( x ) = f ( x ) + 8g(x) =f(x) +8

Use synthetic division to divide. x3−216x−6\frac { x ^ { 3 } - 216 } { x - 6 }x−6x3−216​

Solve the inequality and graph the solution on the real number line x2≤16x ^ { 2 } \leq 16x2≤16

Write the quadratic function f(x) =−x2+2x+2f ( x ) = - x ^ { 2 } + 2 x + 2f(x) =−x2+2x+2 in standard form.

Perform the operation and write the result in standard form. 41+i−51−i\frac { 4 } { 1 + i } - \frac { 5 } { 1 - i }1+i4​−1−i5​

Select the correct graph of the following function. f(x) =x2−49x+7f ( x ) = \frac { x ^ { 2 } - 49 } { x + 7 }f(x) =x+7x2−49​

If x=7 is a root of x3−3x2−7x+21=0x = \sqrt { 7 } \text { is a root of } x ^ { 3 } - 3 x ^ { 2 } - 7 x + 21 = 0x=7​ is a root of x3−3x2−7x+21=0 , use synthetic division to factor the polynomial completely and list all real solutions of the equation.

Find all the zeros of the function and write the polynomial as a product of linear fac- tors. y4−2401y ^ { 4 } - 2401y4−2401

The total revenue RRR earned per day (in dollars) from a pet-sitting service is given by R(p) =−12p2+150pR ( p ) = - 12 p ^ { 2 } + 150 pR(p) =−12p2+150p where, ppp is the price charged per pet (in dollars) . Find the revenue when the price per pet is $4\$ 4$4 .

Simplify the complex number and write it in standard form. 1(6i) 3\frac { 1 } { ( 6 i ) ^ { 3 } }(6i) 31​

Identify all intercepts of f(x) =x2x2+36f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 36 }f(x) =x2+36x2​

The cost C (in millions of dollars) removing p% the industrial and municipal pollu- tants discharged into a river is given by C=245p100−p,0≤p≤100C = \frac { 245 p } { 100 - p } , 0 \leq p \leq 100C=100−p245p​,0≤p≤100 Select the correct graph of the cost function.

Use long division to divide. (x4+9x3+20x2−x−5) ÷(x+5) \left( x ^ { 4 } + 9 x ^ { 3 } + 20 x ^ { 2 } - x - 5 \right) \div ( x + 5 ) (x4+9x3+20x2−x−5) ÷(x+5)

Find all the rational zeros of the function. x3−15x2+75x−125x ^ { 3 } - 15 x ^ { 2 } + 75 x - 125x3−15x2+75x−125

Use long division to divide. (x3+5x2+16x+80) ÷(x+5) \left( x ^ { 3 } + 5 x ^ { 2 } + 16 x + 80 \right) \div ( x + 5 ) (x3+5x2+16x+80) ÷(x+5)

Perform the operation and write the result in standard form. (10−9i) 2( 10 - 9 i ) ^ { 2 }(10−9i) 2

Find all the rational zeros of the function f(x) =2x5+3x4−9x3−3x2−11x−6f ( x ) = 2 x ^ { 5 } + 3 x ^ { 4 } - 9 x ^ { 3 } - 3 x ^ { 2 } - 11 x - 6f(x) =2x5+3x4−9x3−3x2−11x−6

Solve the inequality and write the solution set in interval notation. x4(x−3) ≤0x ^ { 4 } ( x - 3 ) \leq 0x4(x−3) ≤0