Dividing Polynomials Worksheets - Math Monks - Free Printable

Let’s solve each polynomial long division problem one by one. I’ll walk through the steps clearly for each.

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Problem 2:

> Divide \( 5x^3 - 13x^2 + 10x - 8 \) by \( x - 2 \)

Step 1: Divide leading term of dividend by leading term of divisor:
\[
\frac{5x^3}{x} = 5x^2
\]
Write \( 5x^2 \) above the division bar.

Step 2: Multiply \( 5x^2 \cdot (x - 2) = 5x^3 - 10x^2 \)

Step 3: Subtract:
\[
(5x^3 - 13x^2) - (5x^3 - 10x^2) = -3x^2
\]
Bring down next term: \( -3x^2 + 10x \)

Step 4: Divide \( \frac{-3x^2}{x} = -3x \)

Multiply: \( -3x \cdot (x - 2) = -3x^2 + 6x \)

Subtract:
\[
(-3x^2 + 10x) - (-3x^2 + 6x) = 4x
\]
Bring down next term: \( 4x - 8 \)

Step 5: Divide \( \frac{4x}{x} = 4 \)

Multiply: \( 4 \cdot (x - 2) = 4x - 8 \)

Subtract:
\[
(4x - 8) - (4x - 8) = 0
\]

✔ Answer: \( 5x^2 - 3x + 4 \)

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Problem 3:

> Divide \( x^4 + 4x^3 + x - 10 \) by \( x^2 + 3x - 5 \)

Note: The dividend is missing an \( x^2 \) term → write as \( x^4 + 4x^3 + 0x^2 + x - 10 \)

Step 1: \( \frac{x^4}{x^2} = x^2 \)

Multiply: \( x^2 \cdot (x^2 + 3x - 5) = x^4 + 3x^3 - 5x^2 \)

Subtract:
\[
(x^4 + 4x^3 + 0x^2) - (x^4 + 3x^3 - 5x^2) = x^3 + 5x^2
\]
Bring down next terms: \( x^3 + 5x^2 + x \)

Step 2: \( \frac{x^3}{x^2} = x \)

Multiply: \( x \cdot (x^2 + 3x - 5) = x^3 + 3x^2 - 5x \)

Subtract:
\[
(x^3 + 5x^2 + x) - (x^3 + 3x^2 - 5x) = 2x^2 + 6x
\]
Bring down next term: \( 2x^2 + 6x - 10 \)

Step 3: \( \frac{2x^2}{x^2} = 2 \)

Multiply: \( 2 \cdot (x^2 + 3x - 5) = 2x^2 + 6x - 10 \)

Subtract:
\[
(2x^2 + 6x - 10) - (2x^2 + 6x - 10) = 0
\]

✔ Answer: \( x^2 + x + 2 \)

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Problem 4:

> Divide \( x^4 - 3x^3 + 27x - 81 \) by \( x - 3 \)

Note: Missing \( x^2 \) term → write as \( x^4 - 3x^3 + 0x^2 + 27x - 81 \)

Step 1: \( \frac{x^4}{x} = x^3 \)

Multiply: \( x^3 \cdot (x - 3) = x^4 - 3x^3 \)

Subtract:
\[
(x^4 - 3x^3) - (x^4 - 3x^3) = 0
\]
Bring down next terms: \( 0x^2 + 27x \)

Step 2: \( \frac{0x^2}{x} = 0 \), so we move to next term.

Actually, since we have no \( x^2 \) term, we proceed:

Divide \( \frac{0x^2}{x} = 0 \), then bring down 27x → now divide \( \frac{27x}{x} = 27 \)

Wait — let's do it properly:

After subtracting, we have remainder so far: \( 0x^3 + 0x^2 + 27x - 81 \)

Now divide \( \frac{0x^2}{x} = 0 \), so next term in quotient is 0 for \( x^2 \), then:

Divide \( \frac{27x}{x} = 27 \)

Multiply: \( 27 \cdot (x - 3) = 27x - 81 \)

Subtract:
\[
(27x - 81) - (27x - 81) = 0
\]

So quotient is: \( x^3 + 0x^2 + 0x + 27 = x^3 + 27 \)

✔ Answer: \( x^3 + 27 \)

*(Note: This makes sense because \( x^4 - 3x^3 + 27x - 81 = (x - 3)(x^3 + 27) \))*

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Problem 5:

> Divide \( 2x^3 + 15x^2 - 14x - 48 \) by \( x - 2 \)

Step 1: \( \frac{2x^3}{x} = 2x^2 \)

Multiply: \( 2x^2 \cdot (x - 2) = 2x^3 - 4x^2 \)

Subtract:
\[
(2x^3 + 15x^2) - (2x^3 - 4x^2) = 19x^2
\]
Bring down next term: \( 19x^2 - 14x \)

Step 2: \( \frac{19x^2}{x} = 19x \)

Multiply: \( 19x \cdot (x - 2) = 19x^2 - 38x \)

Subtract:
\[
(19x^2 - 14x) - (19x^2 - 38x) = 24x
\]
Bring down next term: \( 24x - 48 \)

Step 3: \( \frac{24x}{x} = 24 \)

Multiply: \( 24 \cdot (x - 2) = 24x - 48 \)

Subtract:
\[
(24x - 48) - (24x - 48) = 0
\]

✔ Answer: \( 2x^2 + 19x + 24 \)

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Problem 6:

> Divide \( 2x^3 - 8x^2 + 9x - 2 \) by \( x - 2 \)

Step 1: \( \frac{2x^3}{x} = 2x^2 \)

Multiply: \( 2x^2 \cdot (x - 2) = 2x^3 - 4x^2 \)

Subtract:
\[
(2x^3 - 8x^2) - (2x^3 - 4x^2) = -4x^2
\]
Bring down next term: \( -4x^2 + 9x \)

Step 2: \( \frac{-4x^2}{x} = -4x \)

Multiply: \( -4x \cdot (x - 2) = -4x^2 + 8x \)

Subtract:
\[
(-4x^2 + 9x) - (-4x^2 + 8x) = x
\]
Bring down next term: \( x - 2 \)

Step 3: \( \frac{x}{x} = 1 \)

Multiply: \( 1 \cdot (x - 2) = x - 2 \)

Subtract:
\[
(x - 2) - (x - 2) = 0
\]

✔ Answer: \( 2x^2 - 4x + 1 \)

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## ✔ Final Answers:

② \( \boxed{5x^2 - 3x + 4} \)
③ \( \boxed{x^2 + x + 2} \)
④ \( \boxed{x^3 + 27} \)
⑤ \( \boxed{2x^2 + 19x + 24} \)
⑥ \( \boxed{2x^2 - 4x + 1} \)

Let me know if you want to see any of these written out in long division format!