For questions $$ 20-23 $$, use your graphing technology to generate a table or a graph of the given function. Identify what type of function it is (quadratic or cubic) and the $$ x $$-intercepts of the function. Write the function as a product of its factors. $$ \begin{array}{ll} ext { 20. } g(x)=-x^{3}-3 x^{2}+6 x+8 & ext { 21. } g(x)=-x^{2}-4 x-4 \ ext { 22. } g(x)=x^{2}-10 x+21 & ext { 23. } q(x)=4 x^{3}-13 x^{2}+3 x\end{array} $$

Question

For questions $$ 20-23 $$, use your graphing technology to generate a table or a graph of the given function. Identify what type of function it is (quadratic or cubic) and the $$ x $$-intercepts of the function. Write the function as a product of its factors. $$ \begin{array}{ll}	ext { 20. } g(x)=-x^{3}-3 x^{2}+6 x+8 & 	ext { 21. } g(x)=-x^{2}-4 x-4 \ 	ext { 22. } g(x)=x^{2}-10 x+21 & 	ext { 23. } q(x)=4 x^{3}-13 x^{2}+3 x\end{array} $$

UpStudy AI · Accepted Answer

### Problem 20

We are given  
$$
g(x) = -x^3 - 3x^2 + 6x + 8.
$$
This is a **cubic** function.

**Step 1. Factor by first factoring out the negative sign:**  
$$
g(x) = -\Bigl(x^3 + 3x^2 - 6x - 8\Bigr).
$$

**Step 2. Find a rational root of the cubic inside.**  
Try $$ x = -1 $$:  
$$
(-1)^3 + 3(-1)^2 - 6(-1) - 8 = -1 + 3 + 6 - 8 = 0.
$$
Thus, $$ x = -1 $$ is a root and $$ x + 1 $$ is a factor.

**Step 3. Divide the cubic by $$ x + 1 $$:**  
Dividing  
$$
x^3 + 3x^2 - 6x - 8 \quad \text{by} \quad x + 1,
$$
we obtain the quadratic  
$$
x^2 + 2x - 8.
$$

**Step 4. Factor the quadratic:**  
Solve  
$$
x^2 + 2x - 8 = 0.
$$
Its discriminant is  
$$
\Delta = (2)^2 - 4(1)(-8) = 4 + 32 = 36.
$$
Thus,  
$$
x = \frac{-2 \pm 6}{2} \quad \Longrightarrow \quad x = 2 \quad \text{or} \quad x = -4.
$$
This quadratic factors as  
$$
(x - 2)(x + 4).
$$

**Step 5. Write the complete factorization:**  
Include the negative sign we factored out:  
$$
g(x) = -\Bigl( (x+1)(x-2)(x+4) \Bigr).
$$

**Step 6. Identify the $$ x $$-intercepts:**  
Set $$ g(x)=0 $$. The zeros occur when  
$$
x + 1 = 0,\quad x - 2 = 0,\quad x + 4 = 0,
$$
giving  
$$
x = -1,\quad x = 2,\quad x = -4.
$$

#### Table of selected values for $$ g(x) $$
  
| $$ x $$ | $$ g(x) $$ |
|:-------:|:-----------:|
| $$-5$$  | $$-(-5+1)(-5-2)(-5+4)= -((-4)(-7)(-1))= - (28)= -28$$ |
| $$-4$$  | $$0$$ |
| $$-3$$  | $$ -((-3+1)(-3-2)(-3+4)) = -((-2)(-5)(1)) = - (10)= -10$$ |
| $$-1$$  | $$0$$ |
| $$0$$   | $$ -((0+1)(0-2)(0+4))= -(1\cdot -2\cdot 4)= 8$$ |
| $$2$$   | $$0$$ |
| $$3$$   | $$-((3+1)(3-2)(3+4))= -(4\cdot1\cdot7)= -28$$ |

---

### Problem 21

We are given  
$$
g(x) = -x^2 - 4x - 4.
$$
This is a **quadratic** function.

**Step 1. Factor out the negative sign:**  
$$
g(x) = -\Bigl(x^2 + 4x + 4\Bigr).
$$

**Step 2. Factor the quadratic inside:**  
Notice that  
$$
x^2 + 4x + 4 = (x + 2)^2.
$$

**Step 3. Write the factorization:**  
$$
g(x) = -(x+2)^2.
$$

**Step 4. Identify the $$ x $$-intercepts:**  
Set $$ g(x)=0 $$:
$$
-(x+2)^2 = 0 \quad \Longrightarrow \quad (x+2)^2 = 0,
$$
so  
$$
x = -2 \quad (\text{a double root}).
$$

---

### Problem 22

We are given  
$$
g(x) = x^2 - 10x + 21.
$$
This is a **quadratic** function.

**Step 1. Factor the quadratic:**  
We look for two numbers that add to $$-10$$ and multiply to $$21$$. Notice that  
$$
-3 + (-7) = -10 \quad \text{and} \quad (-3)(-7)=21.
$$
Thus,  
$$
x^2 - 10x + 21 = (x-3)(x-7).
$$

**Step 2. Identify the $$ x $$-intercepts:**  
Set  
$$
(x-3)(x-7)=0,
$$
giving  
$$
x = 3 \quad \text{or} \quad x = 7.
$$

#### Table of selected values for $$ g(x) $$
  
| $$ x $$ | $$ g(x) $$ |
|:-------:|:-----------:|
| $$0$$   | $$ 0^2 - 10\cdot 0 + 21 = 21$$ |
| $$3$$   | $$ 3^2 - 10\cdot3 + 21 = 9 - 30 + 21 = 0$$ |
| $$5$$   | $$ 25 - 50 + 21 = -4$$ |
| $$7$$   | $$ 49 - 70 + 21 = 0$$ |

---

### Problem 23

We are given  
$$
q(x) = 4x^3 - 13x^2 + 3x.
$$
This is a **cubic** function.

**Step 1. Factor out the common factor $$ x $$:**  
$$
q(x) = x\Bigl(4x^2 - 13x + 3\Bigr).
$$

**Step 2. Factor the quadratic $$ 4x^2 - 13x + 3 $$:**  
Compute its discriminant:  
$$
\Delta = (-13)^2 - 4 \cdot 4 \cdot 3 = 169 - 48 = 121.
$$
Thus, the two roots are  
$$
x = \frac{13 \pm \sqrt{121}}{8} = \frac{13 \pm 11}{8}.
$$
This gives  
$$
x = \frac{24}{8} = 3 \quad \text{and} \quad x = \frac{2}{8} = \frac{1}{4}.
$$
The quadratic factors as  
$$
4x^2 - 13x + 3 = (x-3)(4x-1).
$$

**Step 3. Write the complete factorization:**  
$$
q(x) = x(x-3)(4x-1).
$$

**Step 4. Identify the $$ x $$-intercepts:**  
The zeros come from  
$$
x = 0,\quad x - 3 = 0 \quad (\text{so } x = 3),\quad 4x - 1 = 0 \quad (\text{so } x = \tfrac{1}{4}).
$$

#### Table of selected values for $$ q(x) $$
  
| $$ x $$  | $$ q(x) $$ |
|:--------:|:-----------:|
| $$-1$$   | $$ -1\Bigl((-1)-3\Bigr)\Bigl(4(-1)-1\Bigr) = -1(-4)(-5)= -20$$ |
| $$0$$    | $$ 0 $$ |
| $$0.25$$ | $$ 0.25\bigl(0.25-3\bigr)\bigl(4(0.25)-1\bigr)= 0.25(-2.75)(0)= 0$$ |
| $$3$$    | $$ 3(3-3)(4\cdot3-1)= 3\cdot 0\cdot 11 = 0$$ |
| $$4$$    | $$ 4(4-3)(16-1)= 4\cdot1\cdot15= 60$$ |

---

### Summary of Answers

- **Problem 20:**  
  Function type: **Cubic**  
  Factorization:  
  $$
  g(x) = -(x+1)(x-2)(x+4)
  $$  
  $$ x $$-intercepts: $$ x = -1, \;2, \;-4 $$.

- **Problem 21:**  
  Function type: **Quadratic**  
  Factorization:  
  $$
  g(x) = -(x+2)^2
  $$  
  $$ x $$-intercept: $$ x = -2 $$ (double root).

- **Problem 22:**  
  Function type: **Quadratic**  
  Factorization:  
  $$
  g(x) = (x-3)(x-7)
  $$  
  $$ x $$-intercepts: $$ x = 3, \;7 $$.

- **Problem 23:**  
  Function type: **Cubic**  
  Factorization:  
  $$
  q(x) = x(x-3)(4x-1)
  $$  
  $$ x $$-intercepts: $$ x = 0,\;3, \; \frac{1}{4} $$.
**Problem 20:** - **Function Type:** Cubic - **Factorization:** $$ g(x) = -(x+1)(x-2)(x+4) $$ - **$$ x $$-Intercepts:** $$ x = -1, \;2, \;-4 $$ **Problem 21:** - **Function Type:** Quadratic - **Factorization:** $$ g(x) = -(x+2)^2 $$ - **$$ x $$-Intercept:** $$ x = -2 $$ (double root) **Problem 22:** - **Function Type:** Quadratic - **Factorization:** $$ g(x) = (x-3)(x-7) $$ - **$$ x $$-Intercepts:** $$ x = 3, \;7 $$ **Problem 23:** - **Function Type:** Cubic - **Factorization:** $$ q(x) = x(x-3)(4x-1) $$ - **$$ x $$-Intercepts:** $$ x = 0, \;3, \;\frac{1}{4} $$

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