Free Printable Polynomial Long Division Worksheets - Free Printable

To solve the problems on this worksheet, we will use both long division and synthetic division where applicable. Let's go through each problem step by step.

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Problem 1: \( (r^2 + 6r + 15) \div (r + 5) \)

#### Solution:
We can use synthetic division since the divisor is of the form \( r + 5 \).

1. Write the coefficients of the dividend \( r^2 + 6r + 15 \): \( [1, 6, 15] \).
2. The root corresponding to \( r + 5 = 0 \) is \( r = -5 \).
3. Set up the synthetic division:

\[
\begin{array}{r|rrr}
-5 & 1 & 6 & 15 \\
& & -5 & -5 \\
\hline
& 1 & 1 & 10 \\
\end{array}
\]

- Bring down the first coefficient: \( 1 \).
- Multiply \( -5 \times 1 = -5 \) and add to the next coefficient: \( 6 + (-5) = 1 \).
- Multiply \( -5 \times 1 = -5 \) and add to the next coefficient: \( 15 + (-5) = 10 \).

The quotient is \( r + 1 \) and the remainder is \( 10 \).

Thus, the result is:
\[
(r^2 + 6r + 15) \div (r + 5) = r + 1 + \frac{10}{r + 5}
\]

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Problem 2: \( (r^2 + 10r + 13) \div (r + 7) \)

#### Solution:
Again, use synthetic division with the root \( r = -7 \).

1. Coefficients of the dividend \( r^2 + 10r + 13 \): \( [1, 10, 13] \).
2. Set up the synthetic division:

\[
\begin{array}{r|rrr}
-7 & 1 & 10 & 13 \\
& & -7 & -21 \\
\hline
& 1 & 3 & -8 \\
\end{array}
\]

- Bring down the first coefficient: \( 1 \).
- Multiply \( -7 \times 1 = -7 \) and add to the next coefficient: \( 10 + (-7) = 3 \).
- Multiply \( -7 \times 3 = -21 \) and add to the next coefficient: \( 13 + (-21) = -8 \).

The quotient is \( r + 3 \) and the remainder is \( -8 \).

Thus, the result is:
\[
(r^2 + 10r + 13) \div (r + 7) = r + 3 - \frac{8}{r + 7}
\]

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Problem 3: \( (n^3 - 5n^2 - 33n - 37) \div (n - 9) \)

#### Solution:
Use synthetic division with the root \( n = 9 \).

1. Coefficients of the dividend \( n^3 - 5n^2 - 33n - 37 \): \( [1, -5, -33, -37] \).
2. Set up the synthetic division:

\[
\begin{array}{r|rrrr}
9 & 1 & -5 & -33 & -37 \\
& & 9 & 36 & 27 \\
\hline
& 1 & 4 & 3 & -10 \\
\end{array}
\]

- Bring down the first coefficient: \( 1 \).
- Multiply \( 9 \times 1 = 9 \) and add to the next coefficient: \( -5 + 9 = 4 \).
- Multiply \( 9 \times 4 = 36 \) and add to the next coefficient: \( -33 + 36 = 3 \).
- Multiply \( 9 \times 3 = 27 \) and add to the next coefficient: \( -37 + 27 = -10 \).

The quotient is \( n^2 + 4n + 3 \) and the remainder is \( -10 \).

Thus, the result is:
\[
(n^3 - 5n^2 - 33n - 37) \div (n - 9) = n^2 + 4n + 3 - \frac{10}{n - 9}
\]

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Problem 4: \( (x^3 + 6x^2 - 30x + 102) \div (x + 10) \)

#### Solution:
Use synthetic division with the root \( x = -10 \).

1. Coefficients of the dividend \( x^3 + 6x^2 - 30x + 102 \): \( [1, 6, -30, 102] \).
2. Set up the synthetic division:

\[
\begin{array}{r|rrrr}
-10 & 1 & 6 & -30 & 102 \\
& & -10 & 40 & -100 \\
\hline
& 1 & -4 & 10 & 2 \\
\end{array}
\]

- Bring down the first coefficient: \( 1 \).
- Multiply \( -10 \times 1 = -10 \) and add to the next coefficient: \( 6 + (-10) = -4 \).
- Multiply \( -10 \times (-4) = 40 \) and add to the next coefficient: \( -30 + 40 = 10 \).
- Multiply \( -10 \times 10 = -100 \) and add to the next coefficient: \( 102 + (-100) = 2 \).

The quotient is \( x^2 - 4x + 10 \) and the remainder is \( 2 \).

Thus, the result is:
\[
(x^3 + 6x^2 - 30x + 102) \div (x + 10) = x^2 - 4x + 10 + \frac{2}{x + 10}
\]

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Final Answer for Problems 1-4:

\[
\boxed{
\begin{aligned}
1. & \quad r + 1 + \frac{10}{r + 5} \\
2. & \quad r + 3 - \frac{8}{r + 7} \\
3. & \quad n^2 + 4n + 3 - \frac{10}{n - 9} \\
4. & \quad x^2 - 4x + 10 + \frac{2}{x + 10}
\end{aligned}
}
\]

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If you need solutions for the remaining problems, please let me know!