Use long division to divide the first polynomial by the second.
5x^3 + 6x^2 − 17x + 20,    x + 3

To divide the polynomial \( 5x^3 + 6x^2 − 17x + 20 \) by \( x + 3 \) using long division, follow these steps:

### Step 1: Set up the division
We will divide \( 5x^3 + 6x^2 - 17x + 20 \) by \( x + 3 \).

### Step 2: Divide the first term of the dividend by the first term of the divisor
Divide \( 5x^3 \) by \( x \), which gives \( 5x^2 \).

### Step 3: Multiply the entire divisor by the result
Multiply \( 5x^2 \) by \( x + 3 \), which gives:
\[
5x^2 \cdot (x + 3) = 5x^3 + 15x^2
\]

### Step 4: Subtract the result from the original polynomial
Now subtract \( 5x^3 + 15x^2 \) from the original polynomial \( 5x^3 + 6x^2 - 17x + 20 \):
\[
(5x^3 + 6x^2 - 17x + 20) - (5x^3 + 15x^2) = -9x^2 - 17x + 20
\]

### Step 5: Divide the first term of the new polynomial by the first term of the divisor
Divide \( -9x^2 \) by \( x \), which gives \( -9x \).

### Step 6: Multiply the divisor by the result
Multiply \( -9x \) by \( x + 3 \):
\[
-9x \cdot (x + 3) = -9x^2 - 27x
\]

### Step 7: Subtract the result from the current polynomial
Subtract \( -9x^2 - 27x \) from \( -9x^2 - 17x + 20 \):
\[
(-9x^2 - 17x + 20) - (-9x^2 - 27x) = 10x + 20
\]

### Step 8: Divide the first term of the new polynomial by the first term of the divisor
Divide \( 10x \) by \( x \), which gives \( 10 \).

### Step 9: Multiply the divisor by the result
Multiply \( 10 \) by \( x + 3 \):
\[
10 \cdot (x + 3) = 10x + 30
\]

### Step 10: Subtract the result from the current polynomial
Subtract \( 10x + 30 \) from \( 10x + 20 \):
\[
(10x + 20) - (10x + 30) = -10
\]

### Final Answer:
The quotient is \( 5x^2 - 9x + 10 \) and the remainder is \( -10 \). So, the result of the division is:
\[
5x^2 - 9x + 10 + \frac{-10}{x + 3}
\]

Do you want more details or have any questions?

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Here are five related questions:
1. How do you handle polynomial division with a divisor of higher degree?
2. What is synthetic division, and how does it differ from long division?
3. How do you determine when a polynomial can be factored instead of divided?
4. What are the applications of polynomial division in calculus?
5. How do you find the remainder when dividing a polynomial using the Remainder Theorem?

**Tip**: Always check your subtraction step carefully to avoid small mistakes when performing long division!

Use long division to divide the first polynomial by the second: 5x^3 + 6x^2 − 17x + 20, x + 3

Learn how to divide the polynomial 5x^3 + 6x^2 − 17x + 20 by x + 3 using long division. This detailed step-by-step solution walks through each division and subtraction step, arriving at the quotient 5x^2 - 9x + 10 and remainder -10. Ideal for students in Grades 9-12.

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