Dividing Polynomials Worksheets - Math Monks - Free Printable

Problem: Solve the given polynomial long divisions.

We are tasked with performing polynomial long division for several problems. Let's solve each one step by step.

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#### Problem 2:
Divide \( 5x^3 - 13x^2 + 10x - 8 \) by \( x - 2 \).

Step 1: Set up the division.
\[
\begin{array}{r|rrrr}
x - 2 & 5x^3 & -13x^2 & +10x & -8 \\
\hline
& 5x^2 & -3x & +4 \\
\end{array}
\]

Step 2: Divide the leading term of the dividend by the leading term of the divisor.
\[
\frac{5x^3}{x} = 5x^2
\]
Write \( 5x^2 \) above the division bar.

Step 3: Multiply \( 5x^2 \) by \( x - 2 \).
\[
5x^2 \cdot (x - 2) = 5x^3 - 10x^2
\]
Write this under the dividend and subtract:
\[
\begin{array}{r|rrrr}
x - 2 & 5x^3 & -13x^2 & +10x & -8 \\
\hline
& 5x^2 & & & \\
& -(5x^3 - 10x^2) & & & \\
\hline
& & -3x^2 & +10x & -8 \\
\end{array}
\]

Step 4: Repeat the process with the new polynomial \( -3x^2 + 10x - 8 \).
\[
\frac{-3x^2}{x} = -3x
\]
Write \( -3x \) above the division bar.

Step 5: Multiply \( -3x \) by \( x - 2 \).
\[
-3x \cdot (x - 2) = -3x^2 + 6x
\]
Write this under the new polynomial and subtract:
\[
\begin{array}{r|rrrr}
x - 2 & 5x^3 & -13x^2 & +10x & -8 \\
\hline
& 5x^2 & -3x & & \\
& -(5x^3 - 10x^2) & & & \\
\hline
& & -3x^2 & +10x & -8 \\
& & -(-3x^2 + 6x) & & \\
\hline
& & & 4x & -8 \\
\end{array}
\]

Step 6: Repeat the process with the new polynomial \( 4x - 8 \).
\[
\frac{4x}{x} = 4
\]
Write \( 4 \) above the division bar.

Step 7: Multiply \( 4 \) by \( x - 2 \).
\[
4 \cdot (x - 2) = 4x - 8
\]
Write this under the new polynomial and subtract:
\[
\begin{array}{r|rrrr}
x - 2 & 5x^3 & -13x^2 & +10x & -8 \\
\hline
& 5x^2 & -3x & +4 & \\
& -(5x^3 - 10x^2) & & & \\
\hline
& & -3x^2 & +10x & -8 \\
& & -(-3x^2 + 6x) & & \\
\hline
& & & 4x & -8 \\
& & & -(4x - 8) & \\
\hline
& & & & 0 \\
\end{array}
\]

The quotient is \( 5x^2 - 3x + 4 \) and the remainder is \( 0 \).

Final Answer for Problem 2:
\[
\boxed{5x^2 - 3x + 4}
\]

---

#### Problem 3:
Divide \( x^4 + 4x^3 + x - 10 \) by \( x^2 + 3x - 5 \).

Step 1: Set up the division.
\[
\begin{array}{r|rrrrr}
x^2 + 3x - 5 & x^4 & +4x^3 & +0x^2 & +x & -10 \\
\hline
& x^2 & +x & +6 \\
\end{array}
\]

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

Step 3: Multiply \( x^2 \) by \( x^2 + 3x - 5 \).
\[
x^2 \cdot (x^2 + 3x - 5) = x^4 + 3x^3 - 5x^2
\]
Write this under the dividend and subtract:
\[
\begin{array}{r|rrrrr}
x^2 + 3x - 5 & x^4 & +4x^3 & +0x^2 & +x & -10 \\
\hline
& x^2 & & & & \\
& -(x^4 + 3x^3 - 5x^2) & & & & \\
\hline
& & x^3 & +5x^2 & +x & -10 \\
\end{array}
\]

Step 4: Repeat the process with the new polynomial \( x^3 + 5x^2 + x - 10 \).
\[
\frac{x^3}{x^2} = x
\]
Write \( x \) above the division bar.

Step 5: Multiply \( x \) by \( x^2 + 3x - 5 \).
\[
x \cdot (x^2 + 3x - 5) = x^3 + 3x^2 - 5x
\]
Write this under the new polynomial and subtract:
\[
\begin{array}{r|rrrrr}
x^2 + 3x - 5 & x^4 & +4x^3 & +0x^2 & +x & -10 \\
\hline
& x^2 & +x & & & \\
& -(x^4 + 3x^3 - 5x^2) & & & & \\
\hline
& & x^3 & +5x^2 & +x & -10 \\
& & -(x^3 + 3x^2 - 5x) & & & \\
\hline
& & & 2x^2 & +6x & -10 \\
\end{array}
\]

Step 6: Repeat the process with the new polynomial \( 2x^2 + 6x - 10 \).
\[
\frac{2x^2}{x^2} = 2
\]
Write \( 2 \) above the division bar.

Step 7: Multiply \( 2 \) by \( x^2 + 3x - 5 \).
\[
2 \cdot (x^2 + 3x - 5) = 2x^2 + 6x - 10
\]
Write this under the new polynomial and subtract:
\[
\begin{array}{r|rrrrr}
x^2 + 3x - 5 & x^4 & +4x^3 & +0x^2 & +x & -10 \\
\hline
& x^2 & +x & +2 & \\
& -(x^4 + 3x^3 - 5x^2) & & & & \\
\hline
& & x^3 & +5x^2 & +x & -10 \\
& & -(x^3 + 3x^2 - 5x) & & & \\
\hline
& & & 2x^2 & +6x & -10 \\
& & & -(2x^2 + 6x - 10) & & \\
\hline
& & & & & 0 \\
\end{array}
\]

The quotient is \( x^2 + x + 2 \) and the remainder is \( 0 \).

Final Answer for Problem 3:
\[
\boxed{x^2 + x + 2}
\]

---

#### Problem 4:
Divide \( x^4 - 3x^3 + 27x - 81 \) by \( x - 3 \).

Step 1: Set up the division.
\[
\begin{array}{r|rrrrr}
x - 3 & x^4 & -3x^3 & +0x^2 & +27x & -81 \\
\hline
& x^3 & & & & \\
\end{array}
\]

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

Step 3: Multiply \( x^3 \) by \( x - 3 \).
\[
x^3 \cdot (x - 3) = x^4 - 3x^3
\]
Write this under the dividend and subtract:
\[
\begin{array}{r|rrrrr}
x - 3 & x^4 & -3x^3 & +0x^2 & +27x & -81 \\
\hline
& x^3 & & & & \\
& -(x^4 - 3x^3) & & & & \\
\hline
& & 0 & +0x^2 & +27x & -81 \\
\end{array}
\]

Step 4: Repeat the process with the new polynomial \( 0x^3 + 0x^2 + 27x - 81 \).
\[
\frac{0x^3}{x} = 0
\]
Write \( 0 \) above the division bar.

Step 5: Multiply \( 0 \) by \( x - 3 \).
\[
0 \cdot (x - 3) = 0
\]
Write this under the new polynomial and subtract:
\[
\begin{array}{r|rrrrr}
x - 3 & x^4 & -3x^3 & +0x^2 & +27x & -81 \\
\hline
& x^3 & +0 & & & \\
& -(x^4 - 3x^3) & & & & \\
\hline
& & 0 & +0x^2 & +27x & -81 \\
& & -0 & & & \\
\hline
& & & 0x^2 & +27x & -81 \\
\end{array}
\]

Step 6: Repeat the process with the new polynomial \( 0x^2 + 27x - 81 \).
\[
\frac{0x^2}{x} = 0
\]
Write \( 0 \) above the division bar.

Step 7: Multiply \( 0 \) by \( x - 3 \).
\[
0 \cdot (x - 3) = 0
\]
Write this under the new polynomial and subtract:
\[
\begin{array}{r|rrrrr}
x - 3 & x^4 & -3x^3 & +0x^2 & +27x & -81 \\
\hline
& x^3 & +0 & +0 & & \\
& -(x^4 - 3x^3) & & & & \\
\hline
& & 0 & +0x^2 & +27x & -81 \\
& & -0 & & & \\
\hline
& & & 0x^2 & +27x & -81 \\
& & & -0 & & \\
\hline
& & & & 27x & -81 \\
\end{array}
\]

Step 8: Repeat the process with the new polynomial \( 27x - 81 \).
\[
\frac{27x}{x} = 27
\]
Write \( 27 \) above the division bar.

Step 9: Multiply \( 27 \) by \( x - 3 \).
\[
27 \cdot (x - 3) = 27x - 81
\]
Write this under the new polynomial and subtract:
\[
\begin{array}{r|rrrrr}
x - 3 & x^4 & -3x^3 & +0x^2 & +27x & -81 \\
\hline
& x^3 & +0 & +0 & +27 & \\
& -(x^4 - 3x^3) & & & & \\
\hline
& & 0 & +0x^2 & +27x & -81 \\
& & -0 & & & \\
\hline
& & & 0x^2 & +27x & -81 \\
& & & -0 & & \\
\hline
& & & & 27x & -81 \\
& & & & -(27x - 81) & \\
\hline
& & & & & 0 \\
\end{array}
\]

The quotient is \( x^3 + 27 \) and the remainder is \( 0 \).

Final Answer for Problem 4:
\[
\boxed{x^3 + 27}
\]

---

#### Problem 5:
Divide \( 2x^3 + 15x^2 - 14x - 48 \) by \( x - 2 \).

Step 1: Set up the division.
\[
\begin{array}{r|rrrr}
x - 2 & 2x^3 & +15x^2 & -14x & -48 \\
\hline
& 2x^2 & +19x & +24 \\
\end{array}
\]

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

Step 3: Multiply \( 2x^2 \) by \( x - 2 \).
\[
2x^2 \cdot (x - 2) = 2x^3 - 4x^2
\]
Write this under the dividend and subtract:
\[
\begin{array}{r|rrrr}
x - 2 & 2x^3 & +15x^2 & -14x & -48 \\
\hline
& 2x^2 & & & \\
& -(2x^3 - 4x^2) & & & \\
\hline
& & 19x^2 & -14x & -48 \\
\end{array}
\]

Step 4: Repeat the process with the new polynomial \( 19x^2 - 14x - 48 \).
\[
\frac{19x^2}{x} = 19x
\]
Write \( 19x \) above the division bar.

Step 5: Multiply \( 19x \) by \( x - 2 \).
\[
19x \cdot (x - 2) = 19x^2 - 38x
\]
Write this under the new polynomial and subtract:
\[
\begin{array}{r|rrrr}
x - 2 & 2x^3 & +15x^2 & -14x & -48 \\
\hline
& 2x^2 & +19x & & \\
& -(2x^3 - 4x^2) & & & \\
\hline
& & 19x^2 & -14x & -48 \\
& & -(19x^2 - 38x) & & \\
\hline
& & & 24x & -48 \\
\end{array}
\]

Step 6: Repeat the process with the new polynomial \( 24x - 48 \).
\[
\frac{24x}{x} = 24
\]
Write \( 24 \) above the division bar.

Step 7: Multiply \( 24 \) by \( x - 2 \).
\[
24 \cdot (x - 2) = 24x - 48
\]
Write this under the new polynomial and subtract:
\[
\begin{array}{r|rrrr}
x - 2 & 2x^3 & +15x^2 & -14x & -48 \\
\hline
& 2x^2 & +19x & +24 & \\
& -(2x^3 - 4x^2) & & & \\
\hline
& & 19x^2 & -14x & -48 \\
& & -(19x^2 - 38x) & & \\
\hline
& & & 24x & -48 \\
& & & -(24x - 48) & \\
\hline
& & & & 0 \\
\end{array}
\]

The quotient is \( 2x^2 + 19x + 24 \) and the remainder is \( 0 \).

Final Answer for Problem 5:
\[
\boxed{2x^2 + 19x + 24}
\]

---

#### Problem 6:
Divide \( 2x^3 - 8x^2 + 9x - 2 \) by \( x - 2 \).

Step 1: Set up the division.
\[
\begin{array}{r|rrrr}
x - 2 & 2x^3 & -8x^2 & +9x & -2 \\
\hline
& 2x^2 & -4x & +1 \\
\end{array}
\]

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

Step 3: Multiply \( 2x^2 \) by \( x - 2 \).
\[
2x^2 \cdot (x - 2) = 2x^3 - 4x^2
\]
Write this under the dividend and subtract:
\[
\begin{array}{r|rrrr}
x - 2 & 2x^3 & -8x^2 & +9x & -2 \\
\hline
& 2x^2 & & & \\
& -(2x^3 - 4x^2) & & & \\
\hline
& & -4x^2 & +9x & -2 \\
\end{array}
\]

Step 4: Repeat the process with the new polynomial \( -4x^2 + 9x - 2 \).
\[
\frac{-4x^2}{x} = -4x
\]
Write \( -4x \) above the division bar.

Step 5: Multiply \( -4x \) by \( x - 2 \).
\[
-4x \cdot (x - 2) = -4x^2 + 8x
\]
Write this under the new polynomial and subtract:
\[
\begin{array}{r|rrrr}
x - 2 & 2x^3 & -8x^2 & +9x & -2 \\
\hline
& 2x^2 & -4x & & \\
& -(2x^3 - 4x^2) & & & \\
\hline
& & -4x^2 & +9x & -2 \\
& & -(-4x^2 + 8x) & & \\
\hline
& & & x & -2 \\
\end{array}
\]

Step 6: Repeat the process with the new polynomial \( x - 2 \).
\[
\frac{x}{x} = 1
\]
Write \( 1 \) above the division bar.

Step 7: Multiply \( 1 \) by \( x - 2 \).
\[
1 \cdot (x - 2) = x - 2
\]
Write this under the new polynomial and subtract:
\[
\begin{array}{r|rrrr}
x - 2 & 2x^3 & -8x^2 & +9x & -2 \\
\hline
& 2x^2 & -4x & +1 & \\
& -(2x^3 - 4x^2) & & & \\
\hline
& & -4x^2 & +9x & -2 \\
& & -(-4x^2 + 8x) & & \\
\hline
& & & x & -2 \\
& & & -(x - 2) & \\
\hline
& & & & 0 \\
\end{array}
\]

The quotient is \( 2x^2 - 4x + 1 \) and the remainder is \( 0 \).

Final Answer for Problem 6:
\[
\boxed{2x^2 - 4x + 1}
\]

---

Final Answers:

1. (Given in the problem)
2. \(\boxed{5x^2 - 3x + 4}\)
3. \(\boxed{x^2 + x + 2}\)
4. \(\boxed{x^3 + 27}\)
5. \(\boxed{2x^2 + 19x + 24}\)
6. \(\boxed{2x^2 - 4x + 1}\)