Divide the next polynomials  using synthetic division: 4x³+3x²-9x+2   divided by x+2, then x⁴+3x³+8x²-x-3   divided by x-1

Let's divide the polynomials using **synthetic division**:

### First Division:  
\[
\frac{4x^3 + 3x^2 - 9x + 2}{x + 2}
\]

1. **Set up synthetic division**:  
   The divisor is \(x + 2\), so we take the opposite of the constant term, which is \(-2\).  
   The coefficients of the dividend are: \(4, 3, -9, 2\).

2. **Perform synthetic division**:

\[
\begin{array}{r|rrrr}
-2 & 4 & 3 & -9 & 2 \\
   &   & -8 & 10 & -2 \\
\hline
   & 4 & -5 & 1  & 0 \\
\end{array}
\]

The quotient is: \(4x^2 - 5x + 1\)  
There is no remainder since the remainder is 0.

### Second Division:
\[
\frac{x^4 + 3x^3 + 8x^2 - x - 3}{x - 1}
\]

1. **Set up synthetic division**:  
   The divisor is \(x - 1\), so we take the opposite of the constant term, which is \(1\).  
   The coefficients of the dividend are: \(1, 3, 8, -1, -3\).

2. **Perform synthetic division**:

\[
\begin{array}{r|rrrrr}
1 & 1 & 3 & 8 & -1 & -3 \\
  &   & 1 & 4 & 12 & 11 \\
\hline
  & 1 & 4 & 12 & 11 & 8 \\
\end{array}
\]

The quotient is: \(x^3 + 4x^2 + 12x + 11\)  
The remainder is 8.

### Summary:
1. \(\frac{4x^3 + 3x^2 - 9x + 2}{x + 2} = 4x^2 - 5x + 1\)
2. \(\frac{x^4 + 3x^3 + 8x^2 - x - 3}{x - 1} = x^3 + 4x^2 + 12x + 11\) with a remainder of 8.

Would you like more details on any of these steps or clarifications?

### 5 Related Questions:
1. What is the difference between synthetic division and long division of polynomials?
2. How do you know when synthetic division is appropriate?
3. Can you perform synthetic division with non-linear divisors?
4. How do you interpret the remainder in polynomial division?
5. How does synthetic division relate to the Factor Theorem?

### Tip:
When dividing polynomials, always check the degree of the divisor—synthetic division works when the divisor is of the form \(x - c\).

Divide the next polynomials using synthetic division: 4x³+3x²-9x+2 divided by x+2, then x⁴+3x³+8x²-x-3 divided by x-1

Learn how to divide the polynomials 4x³ + 3x² - 9x + 2 by x+2, and x⁴ + 3x³ + 8x² - x - 3 by x-1 using synthetic division. This solution covers the Factor and Remainder Theorems, perfect for students in Grades 10-12.

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