divide 10x ^ 3 + 41x ^ 2 - 4x - 15 / 5x + 3 using synthetic division

To divide \( 10x^3 + 41x^2 - 4x - 15 \) by \( 5x + 3 \) using synthetic division, we follow these steps:

### Step 1: Set Up the Problem

First, we rewrite the divisor \( 5x + 3 \) in the form \( x - r \). We can factor out a 5 from the divisor to get:

\[
5x + 3 = 5\left(x + \frac{3}{5}\right)
\]

So, we have \( r = -\frac{3}{5} \). This is the value we will use in synthetic division.

### Step 2: Write Down the Coefficients

Write down the coefficients of the dividend \( 10x^3 + 41x^2 - 4x - 15 \):

\[
10, \quad 41, \quad -4, \quad -15
\]

### Step 3: Synthetic Division Process

1. **Bring down the first coefficient:**  
   \[
   10
   \]

2. **Multiply and add:**  
   - Multiply \( 10 \) by \( -\frac{3}{5} \) to get \( -6 \).
   - Add this to the next coefficient \( 41 \):

   \[
   41 + (-6) = 35
   \]

3. **Repeat the process:**  
   - Multiply \( 35 \) by \( -\frac{3}{5} \) to get \( -21 \).
   - Add this to the next coefficient \( -4 \):

   \[
   -4 + (-21) = -25
   \]

   - Multiply \( -25 \) by \( -\frac{3}{5} \) to get \( 15 \).
   - Add this to the next coefficient \( -15 \):

   \[
   -15 + 15 = 0
   \]

### Step 4: Write Down the Quotient and Remainder

The synthetic division process yields the coefficients of the quotient and any remainder:

- The quotient is \( 10x^2 + 35x - 25 \).
- The remainder is \( 0 \).

### Final Answer:
The result of the division is:

\[
\frac{10x^3 + 41x^2 - 4x - 15}{5x + 3} = 2x^2 + 7x - 5
\]

with a remainder of \( 0 \).

Would you like further details on synthetic division or have any questions? Here are some related questions:

1. How do you perform synthetic division with non-zero remainders?
2. Can synthetic division be used for divisors of higher degrees?
3. What is the difference between synthetic and long division in polynomials?
4. How does synthetic division apply to finding roots of polynomials?
5. What happens in synthetic division if the divisor has a leading coefficient other than 1?

**Tip:** Synthetic division is efficient for dividing polynomials by linear divisors and can also help quickly evaluate polynomials at specific points.

Learn how to divide the polynomial 10x^3 + 41x^2 - 4x - 15 by the linear polynomial 5x + 3 using synthetic division. Follow a detailed step-by-step solution with explanations. Suitable for high school students studying polynomial division and synthetic division.

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