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The problem presented is to find all the zeros of the function \( f(x) = 6x^3 - 53x^2 - 14x + 45 \) given that \( f(9) = 0 \), indicating that \( x = 9 \) is one of the zeros of the polynomial.

To solve this algebraically using synthetic division:

1. **Set up synthetic division** with \( x = 9 \) as the divisor.
2. Write the coefficients of the polynomial: \( 6, -53, -14, 45 \).
3. Perform synthetic division to factor the polynomial into a quotient (a quadratic) and a remainder of 0.
4. Solve the resulting quadratic equation either by factoring, completing the square, or using the quadratic formula to find the other zeros.

Let's go step by step:

### Step 1: Synthetic Division

We set up the synthetic division with \( x = 9 \) as the divisor. The coefficients of the polynomial are \( 6, -53, -14, 45 \).

```
9 |  6   -53   -14   45
```

Perform the division:

1. Bring down the 6.
2. Multiply \( 6 \times 9 = 54 \) and add to -53: \( -53 + 54 = 1 \).
3. Multiply \( 1 \times 9 = 9 \) and add to -14: \( -14 + 9 = -5 \).
4. Multiply \( -5 \times 9 = -45 \) and add to 45: \( 45 + (-45) = 0 \).

So, the result is:

```
  6   1   -5   0
```

This gives us a quotient of \( 6x^2 + x - 5 \) and a remainder of 0, confirming that \( x = 9 \) is indeed a root.

### Step 2: Solve the Quadratic

Now, we solve \( 6x^2 + x - 5 = 0 \). Using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 6 \), \( b = 1 \), and \( c = -5 \).

\[
x = \frac{-1 \pm \sqrt{1^2 - 4(6)(-5)}}{2(6)}
\]
\[
x = \frac{-1 \pm \sqrt{1 + 120}}{12}
\]
\[
x = \frac{-1 \pm \sqrt{121}}{12}
\]
\[
x = \frac{-1 \pm 11}{12}
\]

Thus, the solutions are:

\[
x = \frac{-1 + 11}{12} = \frac{10}{12} = \frac{5}{6}
\]
\[
x = \frac{-1 - 11}{12} = \frac{-12}{12} = -1
\]

### Final Answer:
The zeros of the polynomial \( f(x) \) are:
\[
x = 9, \quad x = \frac{5}{6}, \quad x = -1
\]

Would you like more details or have any questions?

Here are five related questions to deepen understanding:

1. How is synthetic division used to find the zeros of a polynomial?
2. Why does the quadratic formula work for any quadratic equation?
3. Can you perform synthetic division with complex numbers as roots?
4. What are the conditions for a polynomial to have real versus complex roots?
5. How can you check whether a given number is a zero of a polynomial without division?

**Tip:** Always verify your results by substituting the zeros back into the original polynomial to ensure the function equals zero at those points.

If f(x) = 6x^3 - 53x^2 - 14x + 45 and f(9) = 0, then find all of the zeros of f(x) algebraically.

Learn how to find all zeros of the polynomial f(x) = 6x^3 - 53x^2 - 14x + 45 using synthetic division and the quadratic formula. This step-by-step guide covers essential algebraic techniques suitable for high school students in Grades 10-12.

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