Deviding using polynomials using long division and Synthetic Division

(7n4 - 68n3 + 46n2 -7n -18 devide by (n - 9)

We are tasked with dividing the polynomial \( 7n^4 - 68n^3 + 46n^2 - 7n - 18 \) by \( n - 9 \) using both **long division** and **synthetic division**.

### 1. Long Division:

We perform polynomial long division by following these steps:

#### Step 1: Divide the first term of the dividend by the first term of the divisor.

\[
\frac{7n^4}{n} = 7n^3
\]

#### Step 2: Multiply \( 7n^3 \) by \( n - 9 \).

\[
7n^3 \times (n - 9) = 7n^4 - 63n^3
\]

#### Step 3: Subtract the result from the original polynomial.

\[
(7n^4 - 68n^3 + 46n^2 - 7n - 18) - (7n^4 - 63n^3) = -5n^3 + 46n^2 - 7n - 18
\]

#### Step 4: Divide the first term of the new polynomial by \( n \).

\[
\frac{-5n^3}{n} = -5n^2
\]

#### Step 5: Multiply \( -5n^2 \) by \( n - 9 \).

\[
-5n^2 \times (n - 9) = -5n^3 + 45n^2
\]

#### Step 6: Subtract.

\[
(-5n^3 + 46n^2 - 7n - 18) - (-5n^3 + 45n^2) = n^2 - 7n - 18
\]

#### Step 7: Divide \( n^2 \) by \( n \).

\[
\frac{n^2}{n} = n
\]

#### Step 8: Multiply \( n \) by \( n - 9 \).

\[
n \times (n - 9) = n^2 - 9n
\]

#### Step 9: Subtract.

\[
(n^2 - 7n - 18) - (n^2 - 9n) = 2n - 18
\]

#### Step 10: Divide \( 2n \) by \( n \).

\[
\frac{2n}{n} = 2
\]

#### Step 11: Multiply \( 2 \) by \( n - 9 \).

\[
2 \times (n - 9) = 2n - 18
\]

#### Step 12: Subtract.

\[
(2n - 18) - (2n - 18) = 0
\]

Thus, the quotient is \( 7n^3 - 5n^2 + n + 2 \) and the remainder is \( 0 \).

### 2. Synthetic Division:

Synthetic division works well when dividing by linear factors like \( n - 9 \). We will follow these steps:

#### Step 1: Set up the synthetic division.

We use the coefficient of \( n \), which is \( 9 \), and arrange the coefficients of the dividend \( 7n^4 - 68n^3 + 46n^2 - 7n - 18 \).

The coefficients are: \( [7, -68, 46, -7, -18] \).

#### Step 2: Perform synthetic division.

|   9  |   7   |  -68  |   46   |   -7   |  -18  |
|------|-------|-------|--------|--------|-------|
|      |       |  63   |  -45   |     9  |   18  |
|______|_______|_______|________|________|_______|
|      |   7   |   -5  |    1   |     2  |    0  |

#### Step 3: Interpret the result.

The quotient is \( 7n^3 - 5n^2 + n + 2 \) and the remainder is \( 0 \).

### Conclusion:
The quotient from both long division and synthetic division is:

\[
7n^3 - 5n^2 + n + 2
\]

Do you have any questions or want further details on any part of the process?

---

### Related Questions:
1. How do you perform long division with polynomials involving higher-degree terms?
2. Can synthetic division be used if the divisor isn't linear, like \( n^2 + n - 3 \)?
3. How does synthetic division differ from the Ruffini method?
4. What are common mistakes when performing polynomial division?
5. How would you check the quotient and remainder after dividing polynomials?

**Tip:** Synthetic division is most efficient when dividing by a linear factor of the form \( x - a \).

Dividing using polynomials using long division and synthetic division: (7n^4 - 68n^3 + 46n^2 -7n -18) divided by (n - 9).

Learn how to divide the polynomial 7n^4 - 68n^3 + 46n^2 - 7n - 18 by n - 9 using both long division and synthetic division methods. Follow a detailed step-by-step solution, covering key algebraic concepts, suitable for students in Grades 10-12.

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