2. Write a polynomial given the zeros and one point. Be sure to include: - 2-3 examples - Show every step to solve this problem(explain each step) - Color/Highlight to emphasize important information - a Graph of at least ONE polynomial example(Use Desmos).

Question

UpStudy AI · Accepted Answer

### Example 1

**Problem:**  
Find a polynomial given the zeros $$2$$ and $$-1$$ that passes through the point $$(1, 12)$$.

**Step 1: Write the polynomial in factored form**  
Since the zeros are given, the polynomial can be written as:  
$$
f(x) = a(x-2)(x+1)
$$  
*Here, $$a$$ is a constant that we will determine using the given point.*

**Step 2: Substitute the given point to find $$a$$**  
We use the point $$(1, 12)$$ where $$f(1)=12$$:  
$$
12 = a(1-2)(1+1)
$$  
Calculate the factors:  
$$
1-2 = -1 \quad \text{and} \quad 1+1 = 2
$$  
Thus,  
$$
12 = a(-1)(2) = -2a
$$

**Step 3: Solve for $$a$$**  
$$
-2a = 12 \quad \Longrightarrow \quad a = -6
$$

**Step 4: Write the complete polynomial**  
Substitute $$a = -6$$ back into the factored form:  
$$
f(x) = -6(x-2)(x+1)
$$

**Step 5: Expand (Optional)**  
Expanding the factors first:  
$$
(x-2)(x+1) = x^2 + x - 2x - 2 = x^2 - x - 2
$$  
Thus,  
$$
f(x) = -6(x^2 - x - 2) = -6x^2 + 6x + 12
$$

*Final Answer for Example 1:*   
$$
\boxed{f(x) = -6(x-2)(x+1) \quad \text{or} \quad f(x) = -6x^2 + 6x + 12}
$$

**Graph using Desmos:**  
You can graph the polynomial by entering the following into [Desmos Graphing Calculator](https://www.desmos.com/calculator):

$$
y = -6(x-2)(x+1)
$$

---

### Example 2

**Problem:**  
Find a polynomial with zeros $$-3$$ and $$5$$ that passes through the point $$(0, 30)$$.

**Step 1: Write the polynomial in factored form**  
$$
f(x) = a(x+3)(x-5)
$$  
*Again, $$a$$ is the constant to be determined.*

**Step 2: Substitute the given point into the polynomial**  
For the point $$(0, 30)$$:  
$$
30 = a(0+3)(0-5)
$$  
Calculate the factors:  
$$
0+3 = 3 \quad \text{and} \quad 0-5 = -5
$$  
Thus,  
$$
30 = a(3)(-5) = -15a
$$

**Step 3: Solve for $$a$$**  
$$
-15a = 30 \quad \Longrightarrow \quad a = -2
$$

**Step 4: Write the complete polynomial**  
$$
f(x) = -2(x+3)(x-5)
$$

**Step 5: Expand (Optional)**  
Expanding the factors:  
$$
(x+3)(x-5) = x^2 - 5x + 3x -15 = x^2 - 2x -15
$$  
Thus,  
$$
f(x) = -2(x^2 - 2x -15) = -2x^2 + 4x + 30
$$

*Final Answer for Example 2:*   
$$
\boxed{f(x) = -2(x+3)(x-5) \quad \text{or} \quad f(x) = -2x^2 + 4x + 30}
$$

---

### Example 3 (Polynomial with a Repeated Zero)

**Problem:**  
Find a polynomial with a repeated zero $$1$$ (multiplicity 2) and another zero $$-2$$ that passes through the point $$(0, 8)$$.

**Step 1: Write the polynomial in factored form**  
$$
f(x) = a(x-1)^2(x+2)
$$  
*Note that the zero $$1$$ appears twice, hence the square.*

**Step 2: Substitute the given point into the polynomial**  
For the point $$(0, 8)$$:  
$$
8 = a(0-1)^2(0+2)
$$  
Calculate the factors:  
$$
(0-1)^2 = 1 \quad \text{and} \quad (0+2) = 2
$$  
Thus,  
$$
8 = a \cdot 1 \cdot 2 = 2a
$$

**Step 3: Solve for $$a$$**  
$$
2a = 8 \quad \Longrightarrow \quad a = 4
$$

**Step 4: Write the complete polynomial**  
$$
f(x) = 4(x-1)^2(x+2)
$$

**Step 5: Expand (Optional)**  
First, expand $$(x-1)^2$$:  
$$
(x-1)^2 = x^2 - 2x + 1
$$  
Then multiply with $$(x+2)$$:  
$$
(x^2 - 2x + 1)(x+2) = x^3 + 2x^2 -2x^2 -4x + x + 2 = x^3 - 3x + 2
$$  
Multiply by $$4$$:  
$$
f(x) = 4(x^3 -3x +2) = 4x^3 - 12x + 8
$$

*Final Answer for Example 3:*   
$$
\boxed{f(x) = 4(x-1)^2(x+2) \quad \text{or} \quad f(x) = 4x^3 - 12x + 8}
$$

---

Each example shows how to:

1. **Write the polynomial in factored form using the given zeros.**
2. **Substitute the provided point to determine the constant $$a$$.**
3. **(Optional) Expand the polynomial to standard form.**

You can graph any of these polynomials on Desmos. For instance, to graph Example 1, go to [Desmos Graphing Calculator](https://www.desmos.com/calculator) and enter:
$$
y = -6(x-2)(x+1)
$$
**Example 1:** - **Zeros:** 2 and -1 - **Point:** (1, 12) - **Polynomial:** $$ f(x) = -6(x-2)(x+1) $$ or $$ f(x) = -6x^2 + 6x + 12 $$ - **Graph:** [Desmos Graph](https://www.desmos.com/calculator) **Example 2:** - **Zeros:** -3 and 5 - **Point:** (0, 30) - **Polynomial:** $$ f(x) = -2(x+3)(x-5) $$ or $$ f(x) = -2x^2 + 4x + 30 $$ - **Graph:** [Desmos Graph](https://www.desmos.com/calculator) **Example 3:** - **Zeros:** 1 (repeated twice) and -2 - **Point:** (0, 8) - **Polynomial:** $$ f(x) = 4(x-1)^2(x+2) $$ or $$ f(x) = 4x^3 - 12x + 8 $$ - **Graph:** [Desmos Graph](https://www.desmos.com/calculator)

Write a polynomial given the zeros and one point.
Be sure to include:

2-3 examples

Show every step to solve this problem(explain each step)

Color/Highlight to emphasize important information

a Graph of at least ONE polynomial example(Use Desmos).

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Solución

Example 1

Example 2

Example 3 (Polynomial with a Repeated Zero)

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Write a polynomial given the zeros and one point. Be sure to include: 2-3 examples Show every step to solve this problem(explain each step) Color/Highlight to emphasize important information a Graph of at least ONE polynomial example(Use Desmos).