Question 1

Provide an appropriate response.
-Suppose that a polynomial function $f$ is defined in such a way that $f ( - 4.7 ) = - 3$ and $f ( 5 ) = 1$. What conclusion does the Intermediate Value Theorem allow you to make?
A) There must be a zero between $- 3$ and 1 .
B) There must be two zeros between $- 4.7$ and 5 .
C) There must be a zero between $- 4.7$ and 5 .
D) There is not enough information to draw a conclusion about the zeros of $f$.

Accepted Answer

The Intermediate Value Theorem states that if a function $f$ is continuous on a closed interval $[a, b]$, and $N$ is any number between $f(a)$ and $f(b)$, then there is at least one number $c$ in the interval $(a, b)$ such that $f(c) = N$. Since $f(-4.7) = -3$ and $f(5) = 1$, and the function changes sign between these two points, there must be at least one zero (where $f(c) = 0$, which is between $-3$ and $1$) in the interval $(-4.7, 5)$.

Question 2

Provide an appropriate response.
-Fill in the blanks to complete the statement. A polynomial function of degree 11 has at least $x$-intercept(s) and at most $\quad x$-intercept(s).
A) $11 ; 11$
B) $0 ; 10$
C) $10 ; 11$
D) $1 ; 11$

Accepted Answer

A polynomial of degree 11 can have at most 11 x-intercepts (real roots), according to the Fundamental Theorem of Algebra. However, it can also have complex roots, in which case the number of real x-intercepts (real roots) could be fewer, including possibly none. Therefore, it has at least 1 x-intercept and at most 11 x-intercepts.

Question 3

Provide an appropriate response.
-Suppose that $k , a , b$, and $c$ are real numbers and that the polynomial function $f$ may be expressed in factored form as $f ( x ) = ( x - k ) \left( a x ^ { 2 } + b x + c \right)$. If $b ^ { 2 } - 4 a c = - 57$, determine the number of real and imaginary zeros that $f$ has.
A) 3 real zeros, 0 imaginary zeros
B) 2 real zeros, 2 imaginary zeros
C) 2 real zeros, 1 imaginary zero
D) 1 real zero, 2 imaginary zeros

Accepted Answer

The quadratic formula's discriminant, $b^2 - 4ac$, determines the nature of the roots of a quadratic equation $ax^2 + bx + c = 0$. If the discriminant is negative ($b^2 - 4ac < 0$), the equation has two imaginary (complex) roots. Since $b^2 - 4ac = -57$, the quadratic part of the polynomial $ax^2 + bx + c$ has two imaginary roots. The linear factor $(x - k)$ contributes one real root, making the total one real zero and two imaginary zeros.

Question 4

Provide an appropriate response.
-Suppose that $\mathrm { f }$ is a polynomial function of degree 4 . If $- 5$ and 11 are zeros of $f$ and the graph of $f$ is symmetric with respect to the $y$-axis, write $f ( x )$ in factored form.
A) $f ( x ) = ( x + 5 ) ^ { 2 } ( x - 11 ) ^ { 2 }$
B) $f ( x ) = ( x + 5 ) ( x - 11 ) ( x - 5 ) ( x + 11 )$
C) $f ( x ) = ( x + 5 ) ( x - 11 )$
D) $f ( x ) = ( x - 5 ) ^ { 2 } ( x + 11 ) ^ { 2 }$

Accepted Answer

Since the graph of $f$ is symmetric with respect to the $y$-axis and $-5$ and $11$ are zeros, their symmetric counterparts $5$ and $-11$ must also be zeros. Thus, $f(x)$ must have factors $(x + 5)$, $(x - 11)$, $(x - 5)$, and $(x + 11)$, making it a degree 4 polynomial.

Question 5

Provide an appropriate response.
-Suppose that $a$, $b$, and $c$ are real numbers and that the polynomial function $\mathrm { f }$ may be expressed in factored form as $f ( x ) = ( 13 x - 7 ) ( 14 x + 5 ) \left( a x ^ { 2 } + b x + c \right)$. If $b ^ { 2 } - 4 a c = 169$, determine the number of rational, irrational, and imaginary zeros that $f$ has.
A) 2 rational zeros, 2 irrational zeros, 0 imaginary zeros
B) 4 rational zeros, 0 irrational zeros, 0 imaginary zeros
C) 2 rational zeros, 0 irrational zeros, 2 imaginary zeros
D) 0 rational zeros, 0 irrational zeros, 4 imaginary zeros

Accepted Answer

The answer of Provide an appropriate response.
-Suppose that $a$, $b$,...

Question 6

Provide an appropriate response.
-Is it possible to have a polynomial function whose only zeros are 2 and i? If so, write such a function. If not, explain why.
A) No; the Complex Conjugate Zeros Statement requires that - i must also be a zero.
B) Yes; $f ( x ) = x ^ { 2 } - ( 2 - i ) x - 2 i$ is such a function.
C) No; the Complex Conjugate Zeros Statement requires that $- 2$ and $- \mathrm { i }$ must also be zeros.
D) Yes; $f ( x ) = x ^ { 2 } - ( 2 + i ) x + 2 i$ is such a function.

Accepted Answer

The Complex Conjugate Root Theorem states that if a polynomial has real coefficients and a complex number $a + bi$ is a root, then its complex conjugate $a - bi$ must also be a root. Therefore, if $i$ is a root, $-i$ must also be a root.

Question 7

Solve the problem.
-If a rocket is propelled upward from ground level, its height in meters after $t$ seconds is given by $h = - 9.8 \mathrm { t } ^ { 2 } + 98 \mathrm { t }$. During what interval of time will the rocket be higher than $205.8 \mathrm {~m}$ ?
A) $7 < t < 6$
B) $0 < t < 3$
C) $3 < t < 7$
D) $6 < t < 10$

Accepted Answer

To find when the rocket is higher than 205.8 m, set the height equation greater than 205.8 and solve for $t$: $-9.8t^2 + 98t > 205.8$. Rearranging gives $9.8t^2 - 98t + 205.8 < 0$. Factoring or using the quadratic formula gives the roots as $t = 3$ and $t = 7$. Therefore, the rocket is higher than 205.8 m between 3 and 7 seconds.

Question 8

Provide an appropriate response.
-Sketch the graph of a polynomial function which has a negative $y$-intercept, one local maximum and one local minimum, but has no absolute extrema.

A)

B)

C)

D)

Accepted Answer

The answer of Provide an appropriate response.
-Sketch the graph of...

Question 9

Solve the problem.
-A flare fired from the bottom of a gorge is visible only when the flare is above the rim. If it is fired with an initial velocity of $176 \mathrm { ft } / \mathrm { sec }$, and the gorge is $448 \mathrm { ft }$ deep, during what interval can the flare been?
$$\left( h = - 16 t ^ { 2 } + v _ { 0 } t + h _ { 0 } \right. \text { ) }$$
A) $8 < t < 11$
B) $0 < t < 4$
C) $4 < \mathrm { t } < 7$
D) $12 < \mathrm { t } < 15$

Accepted Answer

The equation for the height of the flare is given by $h = -16t^2 + v_0t + h_0$, where $v_0 = 176 \, \text{ft/sec}$ (initial velocity) and $h_0 = -448 \, \text{ft}$ (initial height, since the gorge is 448 ft deep, we consider it as negative). To find when the flare is above the rim (i.e., when $h > 0$), we set $h = 0$ and solve for $t$:$$0 = -16t^2 + 176t - 448$$Dividing the entire equation by -16 to simplify, we get:$$t^2 - 11t + 28 = 0$$Factoring the quadratic equation, we find:$$(t - 4)(t - 7) = 0$$Thus, $t = 4$ and $t = 7$. This means the flare is above the rim of the gorge between $t = 4$ seconds and $t = 7$ seconds.

Question 10

Solve the problem.
-A retailer knows that $\mathrm { n }$ games can be sold in a month if the price is $30 - 0.2 \mathrm { n }$ dollars per game. If he buys each game for $\$ 14$, and if he wishes to make a profit of at least $\$ 300$ per month on sales of this game, how many games must he sell each month?
A) $30 \leq \mathrm { n } \leq 80$
B) $30 \leq n \leq 50$
C) $25 \leq \mathrm { n } \leq 40$
D) $25 \leq \mathrm { n } \leq 30$

Accepted Answer

The answer of Solve the problem.
-A retailer knows that \(\mathrm...

Question 11

Provide an appropriate response.
-Consider the graph of the polynomial function $P ( x ) = x ^ { 3 } + 5 x ^ { 2 } + 3 x - 9$. What are the linear factors of $P ( x )$ ?

A) $x + 3 , x - 1 , x + 6$
B) $x + 1 , x + 3 , x + 5$
C) $x + 3 , x - 1 , x + 3$
D) $x - 3 , x - 1 , x - 3$

Accepted Answer

The answer of Provide an appropriate response.
-Consider the graph of...

Question 12

Solve the problem.
-A coin is tossed upward with an initial velocity of $16 \mathrm { ft } / \mathrm { sec }$. During what interval of time will the coin be at a height of at least $60 \mathrm { ft }$ ? $\left( \mathrm { h } = - 16 \mathrm { t } ^ { 2 } + \mathrm { v } _ { \mathrm { O } } \mathrm { t } + \mathrm { h } _ { \mathrm { O } } \right.$. $)$
A) $0 \leq t \leq 1$
B) $0 \leq t \leq 3$
C) $2 \leq t \leq 3$
D) $3 \leq t \leq 6$

Accepted Answer

The coin will be at a height of at least 60 ft when $h(t) \ge 60$. So, we solve the inequality $-16t^2 + 16t + 0 \ge 60$. This simplifies to $t^2 - t - 3.75 \le 0$, which factors as $(t - 3)(t + 1.25) \le 0$. So, the coin will be at a height of at least 60 ft when $2 \le t \le 3$.

Question 13

Provide an appropriate response.
-Let $\mathrm { f }$ be a polynomial function with real coefficients. If $- 5,8 , - 7 + 4 i$, and $6 - 5 i$ are zeros of $\mathrm { f }$, what is the smallest possible degree that $\mathrm { f }$ can have?
A) 8
B) 4
C) 5
D) 6

Accepted Answer

For a polynomial with real coefficients, complex roots must occur in conjugate pairs. Thus, the zeros include $-7 - 4i$ and $6 + 5i$ as well. This gives a total of 6 distinct zeros, so the smallest possible degree of the polynomial is 6.

Question 14

Provide an appropriate response.
-Determine the number of real and imaginary solutions of the equation $x ^ { 6 } = 1$.
A) 1 real; 0 imaginary
B) 2 real; 4 imaginary
C) 6 real; 0 imaginary
D) 0 real; 6 imaginary

Accepted Answer

The equation $x^6 = 1$ can be solved by recognizing it as a polynomial equation of degree 6. The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n roots, counting multiplicities, in the complex number system. The equation $x^6 = 1$ has 2 real solutions ($x = 1$ and $x = -1$) and 4 complex (imaginary) solutions, which are the 6th roots of unity excluding the real ones.

Quiz 2: Polynomial, Power, and Rational Functions

Provide an appropriate response. -Suppose that a polynomial function $f$ is defined in such a way that $f ( - 4.7 ) = - 3$ and $f ( 5 ) = 1$ . What conclusion does the Intermediate Value Theorem allow you to make?

Provide an appropriate response. -Fill in the blanks to complete the statement. A polynomial function of degree 11 has at least $x$ -intercept(s) and at most $\quad x$ -intercept(s) .

Provide an appropriate response. -Suppose that $\mathrm { f }$ is a polynomial function of degree 4 . If $- 5$ and 11 are zeros of $f$ and the graph of $f$ is symmetric with respect to the $y$ -axis, write $f ( x )$ in factored form.

Provide an appropriate response. -Is it possible to have a polynomial function whose only zeros are 2 and i? If so, write such a function. If not, explain why.

Solve the problem. -If a rocket is propelled upward from ground level, its height in meters after $t$ seconds is given by $h = - 9.8 \mathrm { t } ^ { 2 } + 98 \mathrm { t }$ . During what interval of time will the rocket be higher than $205.8 \mathrm {~m}$ ?

Provide an appropriate response. -Sketch the graph of a polynomial function which has a negative $y$ -intercept, one local maximum and one local minimum, but has no absolute extrema.

Provide an appropriate response. -Let $\mathrm { f }$ be a polynomial function with real coefficients. If $- 5,8 , - 7 + 4 i$ , and $6 - 5 i$ are zeros of $\mathrm { f }$ , what is the smallest possible degree that $\mathrm { f }$ can have?

Provide an appropriate response. -Determine the number of real and imaginary solutions of the equation $x ^ { 6 } = 1$ .

Quiz 2: Polynomial, Power, and Rational Functions

Provide an appropriate response. -Suppose that a polynomial function fff is defined in such a way that f(−4.7) =−3f ( - 4.7 ) = - 3f(−4.7) =−3 and f(5) =1f ( 5 ) = 1f(5) =1 . What conclusion does the Intermediate Value Theorem allow you to make?

Provide an appropriate response. -Fill in the blanks to complete the statement. A polynomial function of degree 11 has at least xxx -intercept(s) and at most x\quad xx -intercept(s) .

Provide an appropriate response. -Suppose that f\mathrm { f }f is a polynomial function of degree 4 . If −5- 5−5 and 11 are zeros of fff and the graph of fff is symmetric with respect to the yyy -axis, write f(x) f ( x ) f(x) in factored form.

Provide an appropriate response. -Is it possible to have a polynomial function whose only zeros are 2 and i? If so, write such a function. If not, explain why.

Solve the problem. -If a rocket is propelled upward from ground level, its height in meters after ttt seconds is given by h=−9.8t2+98th = - 9.8 \mathrm { t } ^ { 2 } + 98 \mathrm { t }h=−9.8t2+98t . During what interval of time will the rocket be higher than 205.8 m205.8 \mathrm {~m}205.8 m ?

Provide an appropriate response. -Sketch the graph of a polynomial function which has a negative yyy -intercept, one local maximum and one local minimum, but has no absolute extrema.

Provide an appropriate response. -Consider the graph of the polynomial function P(x) =x3+5x2+3x−9P ( x ) = x ^ { 3 } + 5 x ^ { 2 } + 3 x - 9P(x) =x3+5x2+3x−9 . What are the linear factors of P(x) P ( x ) P(x) ?

Provide an appropriate response. -Let f\mathrm { f }f be a polynomial function with real coefficients. If −5,8,−7+4i- 5,8 , - 7 + 4 i−5,8,−7+4i , and 6−5i6 - 5 i6−5i are zeros of f\mathrm { f }f , what is the smallest possible degree that f\mathrm { f }f can have?

Provide an appropriate response. -Determine the number of real and imaginary solutions of the equation x6=1x ^ { 6 } = 1x6=1 .