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.