Dividing Polynomials Worksheets - Math Monks - Free Printable

Let's solve each of the polynomial long division problems step by step. We'll go through problems 2 to 6, since problem 1 is already solved for you.

---

Problem 2:

$$
x - 2 \big| 5x^3 - 13x^2 + 10x - 8
$$

We divide $5x^3 - 13x^2 + 10x - 8$ by $x - 2$ using long division.

#### Step 1: Divide the leading terms:
$$
\frac{5x^3}{x} = 5x^2
$$
Write $5x^2$ on top.

Multiply $5x^2(x - 2) = 5x^3 - 10x^2$

Subtract:
$$
(5x^3 - 13x^2) - (5x^3 - 10x^2) = -3x^2
$$
Bring down the next term: $+10x$

Now we have: $-3x^2 + 10x$

#### Step 2:
$$
\frac{-3x^2}{x} = -3x
$$
Write $-3x$ on top.

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

Subtract:
$$
(-3x^2 + 10x) - (-3x^2 + 6x) = 4x
$$
Bring down $-8$: $4x - 8$

#### Step 3:
$$
\frac{4x}{x} = 4
$$
Write $+4$ on top.

Multiply: $4(x - 2) = 4x - 8$

Subtract:
$$
(4x - 8) - (4x - 8) = 0
$$

✔ Answer:
$$
\boxed{5x^2 - 3x + 4}
$$

---

Problem 3:

$$
x^2 + 3x - 5 \big| x^4 + 4x^3 + x - 10
$$

Divide $x^4 + 4x^3 + 0x^2 + x - 10$ by $x^2 + 3x - 5$

#### Step 1:
$$
\frac{x^4}{x^2} = x^2
$$
Write $x^2$ on top.

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

Subtract:
$$
(x^4 + 4x^3 + 0x^2) - (x^4 + 3x^3 - 5x^2) = x^3 + 5x^2
$$
Bring down $+x$: now we have $x^3 + 5x^2 + x$

#### Step 2:
$$
\frac{x^3}{x^2} = x
$$
Write $+x$ on top.

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

Subtract:
$$
(x^3 + 5x^2 + x) - (x^3 + 3x^2 - 5x) = 2x^2 + 6x
$$
Bring down $-10$: $2x^2 + 6x - 10$

#### Step 3:
$$
\frac{2x^2}{x^2} = 2
$$
Write $+2$ on top.

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

Subtract:
$$
(2x^2 + 6x - 10) - (2x^2 + 6x - 10) = 0
$$

✔ Answer:
$$
\boxed{x^2 + x + 2}
$$

---

Problem 4:

$$
x - 3 \big| x^4 - 3x^3 + 27x - 81
$$

Note: The dividend is $x^4 - 3x^3 + 0x^2 + 27x - 81$

#### Step 1:
$$
\frac{x^4}{x} = x^3
$$
Write $x^3$ on top.

Multiply: $x^3(x - 3) = x^4 - 3x^3$

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

So now: $0x^2 + 27x$

#### Step 2:
$$
\frac{0x^2}{x} = 0
$$
Write $0x^2$ on top? Wait — actually, we can skip this and go to next non-zero term.

But let’s continue carefully.

We have $0x^2 + 27x$, so:
$$
\frac{27x}{x} = 27
$$
Write $+27$ on top.

Multiply: $27(x - 3) = 27x - 81$

Subtract:
$$
(27x - 81) - (27x - 81) = 0
$$

Wait — but we skipped a term. Let’s track:

After subtracting $x^4 - 3x^3$, we are left with $0x^2 + 27x - 81$. So we need to bring down both.

So after first step:
- Subtract: $x^4 - 3x^3$ → remainder: $0x^3 + 0x^2 + 27x - 81$

Now divide $0x^2$ by $x$: 0 → write $0x$? But better to jump to next non-zero.

Actually, we can do:

After $x^3$, we have no $x^2$ or $x$ terms in quotient yet.

So:
- Next term: $\frac{0x^2}{x} = 0$, so add $0x^2$ to quotient.
- Then: $\frac{27x}{x} = 27$, so add $+27$

So quotient is: $x^3 + 0x^2 + 27 = x^3 + 27$

Check: $(x - 3)(x^3 + 27)$

Use distributive property:
$$
= x(x^3 + 27) - 3(x^3 + 27) = x^4 + 27x - 3x^3 - 81 = x^4 - 3x^3 + 27x - 81
$$

Perfect match!

✔ Answer:
$$
\boxed{x^3 + 27}
$$

---

Problem 5:

$$
x - 2 \big| 2x^3 + 15x^2 - 14x - 48
$$

Divide $2x^3 + 15x^2 - 14x - 48$ by $x - 2$

#### Step 1:
$$
\frac{2x^3}{x} = 2x^2
$$
Write $2x^2$ on top.

Multiply: $2x^2(x - 2) = 2x^3 - 4x^2$

Subtract:
$$
(2x^3 + 15x^2) - (2x^3 - 4x^2) = 19x^2
$$
Bring down $-14x$: $19x^2 - 14x$

#### Step 2:
$$
\frac{19x^2}{x} = 19x
$$
Write $+19x$ on top.

Multiply: $19x(x - 2) = 19x^2 - 38x$

Subtract:
$$
(19x^2 - 14x) - (19x^2 - 38x) = 24x
$$
Bring down $-48$: $24x - 48$

#### Step 3:
$$
\frac{24x}{x} = 24
$$
Write $+24$ on top.

Multiply: $24(x - 2) = 24x - 48$

Subtract:
$$
(24x - 48) - (24x - 48) = 0
$$

✔ Answer:
$$
\boxed{2x^2 + 19x + 24}
$$

---

Problem 6:

$$
x - 2 \big| 2x^3 - 8x^2 + 9x - 2
$$

Divide $2x^3 - 8x^2 + 9x - 2$ by $x - 2$

#### Step 1:
$$
\frac{2x^3}{x} = 2x^2
$$
Write $2x^2$ on top.

Multiply: $2x^2(x - 2) = 2x^3 - 4x^2$

Subtract:
$$
(2x^3 - 8x^2) - (2x^3 - 4x^2) = -4x^2
$$
Bring down $+9x$: $-4x^2 + 9x$

#### Step 2:
$$
\frac{-4x^2}{x} = -4x
$$
Write $-4x$ on top.

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

Subtract:
$$
(-4x^2 + 9x) - (-4x^2 + 8x) = x
$$
Bring down $-2$: $x - 2$

#### Step 3:
$$
\frac{x}{x} = 1
$$
Write $+1$ on top.

Multiply: $1(x - 2) = x - 2$

Subtract:
$$
(x - 2) - (x - 2) = 0
$$

✔ Answer:
$$
\boxed{2x^2 - 4x + 1}
$$

---

✔ Final Answers:

1. Already done: $x^3 - 3x^2 + 6x - 4$
2. $5x^2 - 3x + 4$
3. $x^2 + x + 2$
4. $x^3 + 27$
5. $2x^2 + 19x + 24$
6. $2x^2 - 4x + 1$

All divisions result in exact quotients with no remainders.