22. if $f(-8) = 0$ and $f(x) = x^3 - x^2 - 58x + 112$, find all the zeros of $f(x)$ and use them to graph the function.
23. the revenue from streaming music services in the u.s. from 2005 to 2016 can be modeled by $y = 0.26x^5 - 7.48x^4 + 79.20x^3 - 333.33x^2 + 481.68x + 99.13$, where $x$ is the number of years since 2005 and $y$ is the revenue in millions of dollars. what might the revenue from streaming music services have been in 2020? what assumption did you make to make your prediction?
24. marcela evaluates the polynomial $p(x) = x^3 - 5x^2 + 3x + 5$ for a factor. some of her work is shown below. find the values of $a$ and $b$.
\begin{array}{r|rrrr}
a & 1 & -5 & 3 & 5 \\
& & 11 & 66 & 759 \\
\hline
& 1 & 6 & 69 & b \\
\end{array}
25. the polynomial function $p(x)$ is symmetric in the $y$-axis and contains the point $(2, -5)$. what is the remainder when $p(x)$ is divided by $(x + 2)$? explain your reasoning.
26. verify the remainder theorem for the polynomial $x^2 + 3x + 5$ and the factor $x - \sqrt{3}$ by first using synthetic division and then evaluating for $x = \sqrt{3}$.

Question

22. if $f(-8) = 0$ and $f(x) = x^3 - x^2 - 58x + 112$, find all the zeros of $f(x)$ and use them to graph the function.
23. the revenue from streaming music services in the u.s. from 2005 to 2016 can be modeled by $y = 0.26x^5 - 7.48x^4 + 79.20x^3 - 333.33x^2 + 481.68x + 99.13$, where $x$ is the number of years since 2005 and $y$ is the revenue in millions of dollars. what might the revenue from streaming music services have been in 2020? what assumption did you make to make your prediction?
24. marcela evaluates the polynomial $p(x) = x^3 - 5x^2 + 3x + 5$ for a factor. some of her work is shown below. find the values of $a$ and $b$.
\begin{array}{r|rrrr}
 a & 1 & -5 & 3 & 5 \\
 & & 11 & 66 & 759 \\
\hline
 & 1 & 6 & 69 & b \\
\end{array}
25. the polynomial function $p(x)$ is symmetric in the $y$-axis and contains the point $(2, -5)$. what is the remainder when $p(x)$ is divided by $(x + 2)$? explain your reasoning.
26. verify the remainder theorem for the polynomial $x^2 + 3x + 5$ and the factor $x - \sqrt{3}$ by first using synthetic division and then evaluating for $x = \sqrt{3}$.

Accepted Answer

### Question 22
# Explanation:
## Step 1: Identify the factor
Since $ f(-8) = 0 $, by the Factor Theorem, $ (x + 8) $ is a factor of $ f(x)=x^{3}-x^{2}-58x + 112 $.
## Step 2: Perform polynomial division or use synthetic division
We use synthetic division with root $ -8 $:
$$
\begin{array}{r|rrrr}
-8 & 1 & -1 & -58 & 112 \
& & -8 & 72 & -112 \
\hline
& 1 & -9 & 14 & 0 \
\end{array}
$$
So, $ f(x)=(x + 8)(x^{2}-9x + 14) $.
## Step 3: Factor the quadratic
Factor $ x^{2}-9x + 14 $: we need two numbers that multiply to $ 14 $ and add to $ -9 $. The numbers are $ -2 $ and $ -7 $. So, $ x^{2}-9x + 14=(x - 2)(x - 7) $.
## Step 4: Find the zeros
Set $ f(x)=0 $: $ (x + 8)(x - 2)(x - 7)=0 $. The zeros are $ x=-8 $, $ x = 2 $, and $ x = 7 $.
# Answer:
The zeros of $ f(x) $ are $ \boldsymbol{-8} $, $ \boldsymbol{2} $, and $ \boldsymbol{7} $.

### Question 23
# Explanation:
## Step 1: Determine the value of $ x $ for 2020
Since $ x $ is the number of years since 2005, for 2020, $ x=2020 - 2005=15 $.
## Step 2: Substitute $ x = 15 $ into the model
Substitute $ x = 15 $ into $ y = 0.26x^{5}-7.48x^{4}+79.20x^{3}-333.33x^{2}+481.68x + 99.13 $:
$$
\begin{align*}
y&=0.26(15)^{5}-7.48(15)^{4}+79.20(15)^{3}-333.33(15)^{2}+481.68(15)+99.13\
&=0.26(759375)-7.48(50625)+79.20(3375)-333.33(225)+481.68(15)+99.13\
&=197437.5-378675 + 267240-74999.25+7225.2+99.13\
&=(197437.5+267240+7225.2+99.13)-(378675 + 74999.25)\
&=472001.83-453674.25\
&=18327.58
\end{align*}
$$
## Step 3: State the assumption
The assumption is that the revenue model (the given polynomial) continues to hold for the year 2020 (i.e., the trend in revenue from 2005 - 2016 continues to 2020).
# Answer:
The revenue in 2020 might be approximately $ \boldsymbol{18327.58} $ million dollars. The assumption is that the revenue model holds for 2020 (the trend continues).

### Question 24
# Explanation:
## Step 1: Analyze the synthetic division
In synthetic division, the first number in the second row is the product of $ a $ and the leading coefficient (which is $ 1 $). The second number in the third row is $ -5 + a\times1 $. From the table, the second number in the third row is $ 6 $, so:
$ -5 + a=6 $
## Step 2: Solve for $ a $
Solve $ -5 + a=6 $: $ a=6 + 5=11 $.
## Step 3: Solve for $ b $
The last number in the third row $ b $ is the sum of the last number in the first row and the last number in the second row. The last number in the first row is $ 5 $ and the last number in the second row is $ 759 $, so $ b = 5+759 = 764 $.
# Answer:
$ a=\boldsymbol{11} $ and $ b=\boldsymbol{764} $.

### Question 25
# Explanation:
## Step 1: Recall the Remainder Theorem
The Remainder Theorem states that if a polynomial $ P(x) $ is divided by $ (x - c) $, the remainder is $ P(c) $. Here, we are dividing by $ (x + 2)=(x-(-2)) $, so the remainder is $ P(-2) $.
## Step 2: Use the symmetry of the polynomial
Since $ P(x) $ is symmetric about the $ y $-axis, $ P(x) $ is an even function, so $ P(-x)=P(x) $ for all $ x $.
## Step 3: Find $ P(-2) $
We know $ P(2)=-5 $. Since $ P(x) $ is even, $ P(-2)=P(2)=-5 $. So the remainder when $ P(x) $ is divided by $ (x + 2) $ is $ -5 $.
# Answer:
The remainder is $ \boldsymbol{-5} $. The reasoning is: By the Remainder Theorem, the remainder when $ P(x) $ is divided by $ (x + 2) $ is $ P(-2) $. Since $ P(x) $ is symmetric about the $ y $-axis (even function), $ P(-2)=P(2)=-5 $.

### Question 26
# Explanation:
## Step 1: Recall the Remainder Theorem
The Remainder Theorem states that if a polynomial $ f(x) $ is divided by $ (x - c) $, the remainder is $ f(c) $. Here, $ f(x)=x^{2}+3x + 5 $ and $ c=\sqrt{3} $, so we need to show that the remainder from synthetic division is equal to $ f(\sqrt{3}) $.

### Part 1: Synthetic Division
We use synthetic division with root $ \sqrt{3} $ for $ f(x)=x^{2}+3x + 5 $. The coefficients are $ 1 $ (for $ x^{2} $), $ 3 $ (for $ x $), and $ 5 $ (constant term).
$$
\begin{array}{r|rrr}
\sqrt{3} & 1 & 3 & 5 \
& & \sqrt{3} & 3\sqrt{3}+ 3 \
\hline
& 1 & 3+\sqrt{3} & 5 + 3\sqrt{3}+3 \
& & & 8 + 3\sqrt{3} \
\end{array}
$$
Wait, actually, for a quadratic $ ax^{2}+bx + c $, synthetic division with root $ c $ is done as:
Bring down the leading coefficient $ 1 $. Multiply by $ \sqrt{3} $ to get $ \sqrt{3} $, add to the next coefficient $ 3 $ to get $ 3+\sqrt{3} $. Multiply $ 3+\sqrt{3} $ by $ \sqrt{3} $ to get $ 3\sqrt{3}+3 $, add to the constant term $ 5 $ to get $ 5 + 3\sqrt{3}+3=8 + 3\sqrt{3} $. So the remainder from synthetic division is $ 8 + 3\sqrt{3} $.

### Part 2: Evaluate $ f(\sqrt{3}) $
Evaluate $ f(\sqrt{3})=(\sqrt{3})^{2}+3(\sqrt{3})+5 $
$ (\sqrt{3})^{2}=3 $, so $ f(\sqrt{3})=3 + 3\sqrt{3}+5=8 + 3\sqrt{3} $.

Since the remainder from synthetic division and $ f(\sqrt{3}) $ are equal, the Remainder Theorem is verified.
# Answer:
The remainder from synthetic division is $ 8 + 3\sqrt{3} $ and $ f(\sqrt{3})=8 + 3\sqrt{3} $, so the Remainder Theorem is verified.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 22

Step 1: Identify the factor

Step 2: Perform polynomial division or use synthetic division

Step 3: Factor the quadratic

Step 4: Find the zeros

Step 1: Determine the value of \( x \) for 2020

Step 2: Substitute \( x = 15 \) into the model

Step 3: State the assumption

Step 1: Analyze the synthetic division

Step 2: Solve for \( a \)

Step 3: Solve for \( b \)

Answer:

Question 23