I can use the remainder theorem to find the remainder when a polynomial is divided by a binomial and determine if the binomial is a factor of the polynomial. Use the remainder theorem to determine the remainder when $$ \left(x^{2}-2 x+4\right) $$ is divided by $$ (x+3) $$. State the remainder and tell whether $$ (x+3) $$ is a factor or not. Show your work! Remainder: Is $$ (x+3) $$ a factor?

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To use the Remainder Theorem, we need to evaluate the polynomial $$ P(x) = x^2 - 2x + 4 $$ at the value that makes the divisor $$ (x + 3) $$ equal to zero.

1. **Identify the value of $$ x $$**:
   - Set $$ x + 3 = 0 $$ to find the value of $$ x $$:
     $$
     x = -3
     $$

2. **Evaluate the polynomial at $$ x = -3 $$**:
   - Substitute $$ -3 $$ into the polynomial $$ P(x) $$:
     $$
     P(-3) = (-3)^2 - 2(-3) + 4
     $$

3. **Calculate $$ P(-3) $$**:
   - Calculate each term:
     $$
     (-3)^2 = 9
     $$
     $$
     -2(-3) = 6
     $$
   - Now, combine these results:
     $$
     P(-3) = 9 + 6 + 4 = 19
     $$

4. **State the remainder**:
   - The remainder when $$ P(x) $$ is divided by $$ (x + 3) $$ is $$ 19 $$.

5. **Determine if $$ (x + 3) $$ is a factor**:
   - According to the Remainder Theorem, if the remainder is zero, then the binomial is a factor of the polynomial. Since the remainder is $$ 19 $$, which is not zero, we conclude that:
   - $$ (x + 3) $$ is **not** a factor of $$ P(x) $$.

### Summary:
- **Remainder**: $$ 19 $$
- **Is $$ (x + 3) $$ a factor?**: No.
When dividing $$ x^2 - 2x + 4 $$ by $$ x + 3 $$, the remainder is 19. Since the remainder is not zero, $$ x + 3 $$ is not a factor of the polynomial.

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