Use synthetic division to find the result when 2, x, to the power 4 , minus, 14, x, cubed, plus, 28, x, squared, minus, 5, x, minus, 212x 
4
 −14x 
3
 +28x 
2
 −5x−21 is divided by x, minus, 3x−3. If there is a remainder, express the result in the form q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fractionq(x)+ 
b(x)
r(x)
​
 .

To perform synthetic division of the polynomial \( P(x) = 2x^4 - 14x^3 + 28x^2 - 5x - 212 \) by \( x - 3 \), we can follow these steps:

1. **Set up the synthetic division**: We use the zero of \( x - 3 \), which is \( 3 \).

2. **Write down the coefficients of \( P(x) \)**: The coefficients are \( 2, -14, 28, -5, -212 \).

3. **Perform synthetic division**:

   - Write \( 3 \) on the left and the coefficients on the right:
     ```
     3 |  2   -14   28   -5   -212
       |       
     ```
   - Bring down the leading coefficient \( 2 \):
     ```
     3 |  2   -14   28   -5   -212
       |       
       |        6
     ----------------
         2   -8
     ```
   - Multiply \( 3 \) by \( 2 \) (which is \( 6 \)), and add it to \(-14\):
     \(-14 + 6 = -8\).

   - Repeat the process:
     ```
     3 |  2   -14   28   -5   -212
       |        6   -6   66
     -------------------------
         2   -8   22   61
     ```
   - Multiply \( 3 \) by \(-8\) (which is \(-24\)), and add it to \( 28\):
     \( 28 - 24 = 4 \).

   - Multiply \( 3 \) by \( 4 \) (which is \( 12\)), and add it to \(-5\):
     \(-5 + 12 = 7\).

   - Multiply \( 3 \) by \( 7 \) (which is \( 21\)), and add it to \(-212\):
     \(-212 + 21 = -191\).

4. **Final result**: After performing all operations, we find:
   - The quotient \( Q(x) = 2x^3 - 8x^2 + 4x + 7 \).
   - The remainder \( R = -191 \).

Thus, we can express the result of the division as:

\[
P(x) = (x - 3)(2x^3 - 8x^2 + 4x + 7) - 191
\]

So, the division can be expressed in the form:

\[
Q(x) + \frac{R}{x - 3}
\]

where:

\[
Q(x) = 2x^3 - 8x^2 + 4x + 7, \quad R = -191, \quad \text{and } b(x) = x - 3.
\]

### Final Answer:
\[
Q(x) = 2x^3 - 8x^2 + 4x + 7, \quad R = -191
\]

If you have any questions or need further clarification, let me know! Here are some follow-up questions to consider:

1. Would you like to see more examples of synthetic division?
2. Do you want to explore polynomial long division as an alternative method?
3. Would you like to discuss the properties of polynomials further?
4. Are you interested in how synthetic division relates to finding roots of polynomials?
5. Would you like to know more about the Remainder Theorem? 

**Tip**: When performing synthetic division, always double-check your multiplication and addition steps to ensure accuracy.

Use synthetic division to find the result when 2x^4 - 14x^3 + 28x^2 - 5x - 212 is divided by x - 3. If there is a remainder, express the result in the form q(x) + r(x)/b(x).

Explore the synthetic division of the polynomial 2x^4 - 14x^3 + 28x^2 - 5x - 212 by x - 3. This detailed solution reveals the quotient 2x^3 - 8x^2 + 4x + 7 and the remainder -191, providing key insights into polynomial division techniques suitable for Grades 9-12.

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