f(x)=9x4−12x3−38x2−20x−3f(x)=9x4-12x3-38x2-20x-3 has a zero of multiplicty 22 at x=−13x=-13. Find the remaining real zeros by using synthetic division with the known zero, then factor the polynomial completely.

Separate the remaining zeros with a comma. Use fractions, not decimals.

**Remaining** Zeros==

Factored Form: f(x)


We are given the polynomial \( f(x) = 9x^4 - 12x^3 - 38x^2 - 20x - 3 \) and we know that it has a zero of multiplicity 2 at \( x = -\frac{1}{3} \). To find the remaining real zeros, we can use synthetic division with \( x + \frac{1}{3} \) as a divisor.

### Step 1: Synthetic Division

We will perform synthetic division twice, because the zero has a multiplicity of 2, which means we must divide by \( x + \frac{1}{3} \) two times.

1. **First Division:**
    - The coefficients of the polynomial \( 9x^4 - 12x^3 - 38x^2 - 20x - 3 \) are: \( [9, -12, -38, -20, -3] \).
    - We divide by \( x + \frac{1}{3} \) (which corresponds to using \( -\frac{1}{3} \) in synthetic division).

2. **Second Division:**
    - Once we get the result from the first division, we repeat the process to divide again by \( x + \frac{1}{3} \).

Let's perform the division now.After performing synthetic division twice, the quotient we obtain is \( 9x^2 - 18x - 27 \), with a remainder of 0, indicating the division was successful.

### Step 2: Factor the Quotient

Now, we need to factor \( 9x^2 - 18x - 27 \). First, factor out the common factor of 9:

\[
9(x^2 - 2x - 3)
\]

Next, we factor \( x^2 - 2x - 3 \) by finding two numbers that multiply to -3 and add to -2. These numbers are -3 and 1:

\[
9(x - 3)(x + 1)
\]

### Step 3: Factored Form of the Polynomial

The factored form of the original polynomial \( f(x) \) is:

\[
f(x) = 9(x + \frac{1}{3})^2(x - 3)(x + 1)
\]

### Step 4: Remaining Real Zeros

The remaining real zeros of the polynomial are \( x = 3 \) and \( x = -1 \).

### Final Answer:
- **Remaining Zeros**: \( 3, -1 \)
- **Factored Form**: \( f(x) = 9(x + \frac{1}{3})^2(x - 3)(x + 1) \)

Do you need further details on the steps, or have any questions?

Here are some related questions to deepen your understanding:
1. What is synthetic division, and how is it different from long division?
2. How do you determine the multiplicity of a zero in a polynomial?
3. Why is it important that the remainder is 0 when using synthetic division with a known zero?
4. Can you factor a quadratic equation without factoring out a common term first?
5. How does the factored form of a polynomial help in graphing its function?

**Tip**: When factoring polynomials, always check for common factors first to simplify your work.

f(x)=9x^4−12x^3−38x^2−20x−3 has a zero of multiplicity 2 at x=−1/3. Find the remaining real zeros by using synthetic division with the known zero, then factor the polynomial completely. Separate the remaining zeros with a comma. Use fractions, not decimals. Remaining Zeros== Factored Form: f(x)

Discover how to find and factor the real zeros of the polynomial f(x)=9x^4−12x^3−38x^2−20x−3, with a given zero of multiplicity 2 at x=−1/3. Using synthetic division and factoring, this step-by-step solution is ideal for students in Grades 10-12 looking to master polynomial functions.

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