Use the Factor Theorem to determine if $$ x - 3 $$ is a factor of $$ f(x) = x^{3} - 4x^{2} + 5x - 6 $$.

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To determine if $$ x - 3 $$ is a factor of the polynomial $$ f(x) = x^{3} - 4x^{2} + 5x - 6 $$ using the Factor Theorem, we need to evaluate $$ f(3) $$.

According to the Factor Theorem, $$ x - c $$ is a factor of $$ f(x) $$ if and only if $$ f(c) = 0 $$. In this case, $$ c = 3 $$.

Let's calculate $$ f(3) $$:

$$
f(3) = (3)^{3} - 4(3)^{2} + 5(3) - 6
$$

Calculating each term:

1. $$ (3)^{3} = 27 $$
2. $$ 4(3)^{2} = 4 \cdot 9 = 36 $$
3. $$ 5(3) = 15 $$

Now substituting these values into the expression for $$ f(3) $$:

$$
f(3) = 27 - 36 + 15 - 6
$$

Now, simplify step by step:

$$
f(3) = 27 - 36 = -9
$$
$$
f(3) = -9 + 15 = 6
$$
$$
f(3) = 6 - 6 = 0
$$

Since $$ f(3) = 0 $$, we conclude that $$ x - 3 $$ is indeed a factor of $$ f(x) $$.

Thus, the answer is:

$$
\text{Yes, } x - 3 \text{ is a factor of } f(x).
$$
Yes, $$ x - 3 $$ is a factor of $$ f(x) $$.

Use the Factor Theorem to determine if \( x - 3 \) is a factor of \( f(x) = x^{3} - 4x^{2} + 5x - 6 \).

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