Perform the indicated operation. (-42r^4+25r^3-9r^2+12r+42)/(6r^2-r+6)

To perform the division of the polynomial \(-42r^4 + 25r^3 - 9r^2 + 12r + 42\) by \(6r^2 - r + 6\), we will use polynomial long division.

### Step 1: Divide the first term of the dividend by the first term of the divisor
- First term of the dividend: \(-42r^4\)
- First term of the divisor: \(6r^2\)

\[
\text{Quotient term: } \frac{-42r^4}{6r^2} = -7r^2
\]

### Step 2: Multiply the entire divisor by this quotient term
\[
-7r^2 \times (6r^2 - r + 6) = -42r^4 + 7r^3 - 42r^2
\]

### Step 3: Subtract this result from the original polynomial
\[
(-42r^4 + 25r^3 - 9r^2 + 12r + 42) - (-42r^4 + 7r^3 - 42r^2) 
\]
Simplifying:
\[
0r^4 + (25r^3 - 7r^3) + (-9r^2 + 42r^2) + 12r + 42 = 18r^3 + 33r^2 + 12r + 42
\]

### Step 4: Repeat the process with the new polynomial
- First term of the new dividend: \(18r^3\)
- First term of the divisor: \(6r^2\)

\[
\text{Quotient term: } \frac{18r^3}{6r^2} = 3r
\]

### Step 5: Multiply the divisor by the new quotient term
\[
3r \times (6r^2 - r + 6) = 18r^3 - 3r^2 + 18r
\]

### Step 6: Subtract this from the new dividend
\[
(18r^3 + 33r^2 + 12r + 42) - (18r^3 - 3r^2 + 18r)
\]
Simplifying:
\[
0r^3 + (33r^2 + 3r^2) + (12r - 18r) + 42 = 36r^2 - 6r + 42
\]

### Step 7: Repeat the process one more time
- First term of the new dividend: \(36r^2\)
- First term of the divisor: \(6r^2\)

\[
\text{Quotient term: } \frac{36r^2}{6r^2} = 6
\]

### Step 8: Multiply the divisor by the new quotient term
\[
6 \times (6r^2 - r + 6) = 36r^2 - 6r + 36
\]

### Step 9: Subtract this from the current dividend
\[
(36r^2 - 6r + 42) - (36r^2 - 6r + 36)
\]
Simplifying:
\[
0r^2 + 0r + 6 = 6
\]

### Final Result
The quotient is:
\[
-7r^2 + 3r + 6
\]
The remainder is:
\[
6
\]
Thus, the final answer is:
\[
\frac{-42r^4 + 25r^3 - 9r^2 + 12r + 42}{6r^2 - r + 6} = -7r^2 + 3r + 6 + \frac{6}{6r^2 - r + 6}
\]

Would you like more details or have any questions?

Here are five related questions:
1. How do you perform synthetic division for polynomials?
2. What are the steps to factor a quadratic expression?
3. How can you check the result of polynomial division?
4. What is the relationship between polynomial division and the remainder theorem?
5. How do you find the roots of a polynomial after dividing it?

**Tip:** Always double-check your subtraction step in polynomial long division to avoid common errors.

Learn how to divide the polynomial -42r^4 + 25r^3 - 9r^2 + 12r + 42 by 6r^2 - r + 6 using polynomial long division. Follow a step-by-step solution with detailed explanations. Suitable for advanced high school students (Grades 11-12).

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