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Given the following polynomial function, P(x)
, evaluated at x=−2
,What would be the remainder if we were to divide P(x)
by  (x+2)
using synthetic division?

To determine the remainder when dividing the polynomial \( P(x) = -3x^2 + 7x - 5 \) by \( x + 2 \) using synthetic division, follow these steps:

### Step 1: Set up the synthetic division
Since you are dividing by \( x + 2 \), the value of \( x \) used in synthetic division is \( -2 \).

### Step 2: Write down the coefficients of \( P(x) \)
The polynomial \( P(x) = -3x^2 + 7x - 5 \) has coefficients \(-3\), \(7\), and \(-5\).

So, we'll use the coefficients:
\[
-3, \quad 7, \quad -5
\]

### Step 3: Perform synthetic division
1. Bring down the leading coefficient \(-3\).
2. Multiply \(-2\) by \(-3\) to get \(6\).
3. Add \(6\) to \(7\) to get \(13\).
4. Multiply \(-2\) by \(13\) to get \(-26\).
5. Add \(-26\) to \(-5\) to get \(-31\).

The final setup looks like this:

\[
\begin{array}{r|rrr}
-2 & -3 & 7 & -5 \\
   &    & 6 & -26 \\
\hline
   & -3 & 13 & -31 \\
\end{array}
\]

### Step 4: Interpret the result
The last number, \(-31\), is the remainder.

### Conclusion
The remainder when dividing \( P(x) \) by \( x + 2 \) is \(-31\).

Would you like any further details or explanations?

Here are five related questions you might find useful:
1. How do you perform synthetic division with more complex polynomials?
2. What are the general steps to find the remainder using the Remainder Theorem?
3. Can you explain why the remainder of synthetic division matches the evaluation of the polynomial at the divisor's root?
4. How does synthetic division relate to factoring polynomials?
5. What is the significance of the remainder in polynomial division?

**Tip:** Remember that synthetic division is a quick way to find the remainder when dividing polynomials, especially when dealing with linear divisors like \( x - c \).

Discover how to find the remainder of the polynomial P(x) = -3x^2 + 7x - 5 divided by (x + 2) using synthetic division. Follow a detailed step-by-step solution, including coefficients and calculations. Suitable for high school students and learners interested in polynomial division and synthetic division techniques.

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