https://file.aiquickdraw.com/mathbot/file/1730061280875tezmv4gy.jpg

To form a polynomial with the given properties:

### Given Information
- Zeros: 
  - \(6\) with multiplicity \(1\)
  - \(-4\) with multiplicity \(2\)
- Degree: \(3\)
- Leading coefficient: \(1\)

### Solution
1. **Write each factor for the given zeros and their multiplicities**:
   - For zero \(6\) with multiplicity \(1\): the factor is \((x - 6)\).
   - For zero \(-4\) with multiplicity \(2\): the factor is \((x + 4)^2\).

2. **Combine the factors**:
   \[
   f(x) = (x - 6)(x + 4)^2
   \]

3. **Expand the expression**:
   First, expand \((x + 4)^2\):
   \[
   (x + 4)^2 = x^2 + 8x + 16
   \]
   Now multiply by \((x - 6)\):
   \[
   f(x) = (x - 6)(x^2 + 8x + 16)
   \]

4. **Distribute \( (x - 6) \) across each term in \( (x^2 + 8x + 16) \):
   \[
   f(x) = x(x^2 + 8x + 16) - 6(x^2 + 8x + 16)
   \]
   \[
   f(x) = x^3 + 8x^2 + 16x - 6x^2 - 48x - 96
   \]

5. **Combine like terms**:
   \[
   f(x) = x^3 + 2x^2 - 32x - 96
   \]

### Final Answer
The polynomial is:
\[
f(x) = x^3 + 2x^2 - 32x - 96
\]

This polynomial has integer coefficients, a leading coefficient of \(1\), and the correct zeros with specified multiplicities.

Would you like a deeper breakdown of any step?

---

Here are five related questions to deepen your understanding:
1. What is the impact of a zero's multiplicity on the shape of a polynomial's graph?
2. How would the polynomial change if the zero \(-4\) had multiplicity \(1\) instead of \(2\)?
3. How can you confirm the degree of the polynomial after expansion?
4. How does the leading coefficient affect the end behavior of the polynomial?
5. What method can you use to factor polynomials when only given some of the roots?

**Tip**: When expanding polynomials, keep track of each term carefully to avoid errors in sign or coefficient.

Form a polynomial whose zeros and degree are given.
Zeros: 6, multiplicity 1; -4, multiplicity 2; degree 3. Type a polynomial with integer coefficients and a leading coefficient of 1.

Learn how to form a polynomial with specific zeros, multiplicities, and a leading coefficient of 1. This problem covers key algebraic concepts, including polynomial expansion and the Fundamental Theorem of Algebra. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation