Find the remainder when ( 4x^5-3x^3+4x^2+2x-12) is divided by (x+3) using synthetic division

We can use synthetic division to find the remainder when \( 4x^5 - 3x^3 + 4x^2 + 2x - 12 \) is divided by \( x + 3 \). In synthetic division, we set up the coefficients of the polynomial and divide by the opposite of the divisor's constant term.

### Step 1: Setup
- The divisor is \( x + 3 \), so we use \( -3 \) in the synthetic division.
- The coefficients of the dividend polynomial \( 4x^5 - 3x^3 + 4x^2 + 2x - 12 \) are:
  - \( 4x^5 \) has coefficient \( 4 \)
  - There's no \( x^4 \) term, so the coefficient is \( 0 \)
  - \( -3x^3 \) has coefficient \( -3 \)
  - \( 4x^2 \) has coefficient \( 4 \)
  - \( 2x \) has coefficient \( 2 \)
  - The constant term \( -12 \) has coefficient \( -12 \)

Thus, the coefficients are: \( [4, 0, -3, 4, 2, -12] \).

### Step 2: Perform synthetic division
We now perform synthetic division with \( -3 \).

\[
\begin{array}{r|rrrrrr}
-3 & 4 & 0 & -3 & 4 & 2 & -12 \\
   &   & -12 & 36 & -99 & 285 & -859 \\
\hline
   & 4 & -12 & 33 & -95 & 287 & -871 \\
\end{array}
\]

### Step 3: Interpretation of the result
- The last number \( -871 \) is the remainder.
- The quotient is \( 4x^4 - 12x^3 + 33x^2 - 95x + 287 \).

### Final Answer:
The remainder when \( 4x^5 - 3x^3 + 4x^2 + 2x - 12 \) is divided by \( x + 3 \) is \( \boxed{-871} \).

Would you like a breakdown of each step or have any questions? 

Here are some related questions to explore further:
1. How does synthetic division differ from polynomial long division?
2. What would the quotient look like if we divided a different polynomial by \( x + 3 \)?
3. How does the Remainder Theorem relate to synthetic division?
4. What happens if the divisor is of a higher degree, such as \( x^2 + 3x + 1 \)?
5. Can you apply synthetic division when dividing by \( x - 3 \)?

**Tip**: When using synthetic division, always ensure that any missing powers of \( x \) are accounted for by inserting zeros as placeholders.

Find the remainder when (4x^5 - 3x^3 + 4x^2 + 2x - 12) is divided by (x + 3) using synthetic division.

Explore how to find the remainder when dividing the polynomial 4x^5 - 3x^3 + 4x^2 + 2x - 12 by (x + 3) using synthetic division. Learn the step-by-step process, including identifying coefficients and applying the Remainder Theorem. Suitable for Grades 9-12 students.

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