Free Printable Polynomial Long Division Worksheets - Free Printable

Problem: Dividing Polynomials Using Long Division

The task is to divide the given polynomials using long division. Below, I will solve each problem step by step.

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#### 1. \( (k^3 + 8k^2 + 10k + 21) \div (k + 7) \)

Step 1: Set up the long division.
\[
\begin{array}{r|rrrr}
k + 7 & k^3 & +8k^2 & +10k & +21 \\
\end{array}
\]

Step 2: Divide the leading term of the dividend by the leading term of the divisor.
\[
\frac{k^3}{k} = k^2
\]
Write \( k^2 \) above the division bar.

Step 3: Multiply \( k^2 \) by the divisor \( k + 7 \).
\[
k^2 \cdot (k + 7) = k^3 + 7k^2
\]
Write this product under the dividend and subtract:
\[
\begin{array}{r|rrrr}
k + 7 & k^3 & +8k^2 & +10k & +21 \\
& -(k^3 & +7k^2) & & \\
\hline
& 0 & +k^2 & +10k & +21 \\
\end{array}
\]

Step 4: Repeat the process with the new polynomial \( k^2 + 10k + 21 \).
Divide the leading term \( k^2 \) by \( k \):
\[
\frac{k^2}{k} = k
\]
Write \( k \) above the division bar.

Multiply \( k \) by the divisor \( k + 7 \):
\[
k \cdot (k + 7) = k^2 + 7k
\]
Subtract this from the current polynomial:
\[
\begin{array}{r|rrrr}
k + 7 & k^3 & +8k^2 & +10k & +21 \\
& -(k^3 & +7k^2) & & \\
\hline
& 0 & +k^2 & +10k & +21 \\
& & -(k^2 & +7k) & \\
\hline
& & 0 & +3k & +21 \\
\end{array}
\]

Step 5: Repeat the process with the new polynomial \( 3k + 21 \).
Divide the leading term \( 3k \) by \( k \):
\[
\frac{3k}{k} = 3
\]
Write \( 3 \) above the division bar.

Multiply \( 3 \) by the divisor \( k + 7 \):
\[
3 \cdot (k + 7) = 3k + 21
\]
Subtract this from the current polynomial:
\[
\begin{array}{r|rrrr}
k + 7 & k^3 & +8k^2 & +10k & +21 \\
& -(k^3 & +7k^2) & & \\
\hline
& 0 & +k^2 & +10k & +21 \\
& & -(k^2 & +7k) & \\
\hline
& & 0 & +3k & +21 \\
& & & -(3k & +21) \\
\hline
& & & 0 & 0 \\
\end{array}
\]

Final Answer:
\[
\boxed{k^2 + k + 3}
\]

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#### 2. \( (n^4 - 17n^3 + 81n^2 - 65n - 56) \div (n - 8) \)

Step 1: Set up the long division.
\[
\begin{array}{r|rrrrr}
n - 8 & n^4 & -17n^3 & +81n^2 & -65n & -56 \\
\end{array}
\]

Step 2: Divide the leading term of the dividend by the leading term of the divisor.
\[
\frac{n^4}{n} = n^3
\]
Write \( n^3 \) above the division bar.

Step 3: Multiply \( n^3 \) by the divisor \( n - 8 \).
\[
n^3 \cdot (n - 8) = n^4 - 8n^3
\]
Write this product under the dividend and subtract:
\[
\begin{array}{r|rrrrr}
n - 8 & n^4 & -17n^3 & +81n^2 & -65n & -56 \\
& -(n^4 & -8n^3) & & & \\
\hline
& 0 & -9n^3 & +81n^2 & -65n & -56 \\
\end{array}
\]

Step 4: Repeat the process with the new polynomial \( -9n^3 + 81n^2 - 65n - 56 \).
Divide the leading term \( -9n^3 \) by \( n \):
\[
\frac{-9n^3}{n} = -9n^2
\]
Write \( -9n^2 \) above the division bar.

Multiply \( -9n^2 \) by the divisor \( n - 8 \):
\[
-9n^2 \cdot (n - 8) = -9n^3 + 72n^2
\]
Subtract this from the current polynomial:
\[
\begin{array}{r|rrrrr}
n - 8 & n^4 & -17n^3 & +81n^2 & -65n & -56 \\
& -(n^4 & -8n^3) & & & \\
\hline
& 0 & -9n^3 & +81n^2 & -65n & -56 \\
& & -(-9n^3 & +72n^2) & & \\
\hline
& & 0 & +9n^2 & -65n & -56 \\
\end{array}
\]

Step 5: Repeat the process with the new polynomial \( 9n^2 - 65n - 56 \).
Divide the leading term \( 9n^2 \) by \( n \):
\[
\frac{9n^2}{n} = 9n
\]
Write \( 9n \) above the division bar.

Multiply \( 9n \) by the divisor \( n - 8 \):
\[
9n \cdot (n - 8) = 9n^2 - 72n
\]
Subtract this from the current polynomial:
\[
\begin{array}{r|rrrrr}
n - 8 & n^4 & -17n^3 & +81n^2 & -65n & -56 \\
& -(n^4 & -8n^3) & & & \\
\hline
& 0 & -9n^3 & +81n^2 & -65n & -56 \\
& & -(-9n^3 & +72n^2) & & \\
\hline
& & 0 & +9n^2 & -65n & -56 \\
& & & -(9n^2 & -72n) & \\
\hline
& & & 0 & +7n & -56 \\
\end{array}
\]

Step 6: Repeat the process with the new polynomial \( 7n - 56 \).
Divide the leading term \( 7n \) by \( n \):
\[
\frac{7n}{n} = 7
\]
Write \( 7 \) above the division bar.

Multiply \( 7 \) by the divisor \( n - 8 \):
\[
7 \cdot (n - 8) = 7n - 56
\]
Subtract this from the current polynomial:
\[
\begin{array}{r|rrrrr}
n - 8 & n^4 & -17n^3 & +81n^2 & -65n & -56 \\
& -(n^4 & -8n^3) & & & \\
\hline
& 0 & -9n^3 & +81n^2 & -65n & -56 \\
& & -(-9n^3 & +72n^2) & & \\
\hline
& & 0 & +9n^2 & -65n & -56 \\
& & & -(9n^2 & -72n) & \\
\hline
& & & 0 & +7n & -56 \\
& & & & -(7n & -56) \\
\hline
& & & & 0 & 0 \\
\end{array}
\]

Final Answer:
\[
\boxed{n^3 - 9n^2 + 9n + 7}
\]

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Continue solving the remaining problems similarly. If you need further solutions, let me know!