Divide Polynomials Worksheet-1 Worksheets - Free Printable

Let's solve each of the polynomial division problems step by step. I'll go through each one, showing the division process and simplifying where necessary.

---

1.) $(24x^4 + 24x^3 + 3x^2) \div 8x^2$

We divide each term in the numerator by $8x^2$:

$$
\frac{24x^4}{8x^2} = 3x^2,\quad
\frac{24x^3}{8x^2} = 3x,\quad
\frac{3x^2}{8x^2} = \frac{3}{8}
$$

So the result is:
$$
3x^2 + 3x + \frac{3}{8}
$$

✔ Answer: $3x^2 + 3x + \frac{3}{8}$

---

2.) $(16x^5 + 2x^4 + x^3) \div 4x$

Divide each term by $4x$:

$$
\frac{16x^5}{4x} = 4x^4,\quad
\frac{2x^4}{4x} = \frac{1}{2}x^3,\quad
\frac{x^3}{4x} = \frac{1}{4}x^2
$$

✔ Answer: $4x^4 + \frac{1}{2}x^3 + \frac{1}{4}x^2$

---

3.) $(27x^3 + 9x^2 + 9x^4) \div 9x^2$

First, arrange in descending powers: $9x^4 + 27x^3 + 9x^2$

Now divide each term by $9x^2$:

$$
\frac{9x^4}{9x^2} = x^2,\quad
\frac{27x^3}{9x^2} = 3x,\quad
\frac{9x^2}{9x^2} = 1
$$

✔ Answer: $x^2 + 3x + 1$

---

4.) $(2x^3 - 8x^2 + 12x^4) \div 4x^2$

Arrange in descending order: $12x^4 + 2x^3 - 8x^2$

Divide each term by $4x^2$:

$$
\frac{12x^4}{4x^2} = 3x^2,\quad
\frac{2x^3}{4x^2} = \frac{1}{2}x,\quad
\frac{-8x^2}{4x^2} = -2
$$

✔ Answer: $3x^2 + \frac{1}{2}x - 2$

---

5.) $(x^2 + 9x + 17) \div (x - 4)$

Use polynomial long division:

Divide $x^2$ by $x$: get $x$

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

Subtract:
$(x^2 + 9x + 17) - (x^2 - 4x) = 13x + 17$

Now divide $13x$ by $x$: get $13$

Multiply: $13(x - 4) = 13x - 52$

Subtract:
$(13x + 17) - (13x - 52) = 69$

So quotient is $x + 13$, remainder $69$

✔ Answer: $x + 13 + \frac{69}{x - 4}$

---

6.) $(x^2 + x - 16) \div (x - 4)$

Long division:

Divide $x^2$ by $x$: get $x$

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

Subtract:
$(x^2 + x - 16) - (x^2 - 4x) = 5x - 16$

Divide $5x$ by $x$: get $5$

Multiply: $5(x - 4) = 5x - 20$

Subtract:
$(5x - 16) - (5x - 20) = 4$

✔ Answer: $x + 5 + \frac{4}{x - 4}$

---

7.) $(x^2 - 6x + 2) \div (x - 1)$

Long division:

Divide $x^2$ by $x$: get $x$

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

Subtract:
$(x^2 - 6x + 2) - (x^2 - x) = -5x + 2$

Divide $-5x$ by $x$: get $-5$

Multiply: $-5(x - 1) = -5x + 5$

Subtract:
$(-5x + 2) - (-5x + 5) = -3$

✔ Answer: $x - 5 - \frac{3}{x - 1}$

---

8.) $(x^2 - 24) \div (x - 5)$

Note: $x^2 - 24$ is not a perfect square, so we do long division.

Divide $x^2$ by $x$: get $x$

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

Subtract:
$(x^2 - 24) - (x^2 - 5x) = 5x - 24$

Divide $5x$ by $x$: get $5$

Multiply: $5(x - 5) = 5x - 25$

Subtract:
$(5x - 24) - (5x - 25) = 1$

✔ Answer: $x + 5 + \frac{1}{x - 5}$

---

9.) $(2x^2 - 8x + 2) \div (x - 3)$

Long division:

Divide $2x^2$ by $x$: get $2x$

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

Subtract:
$(2x^2 - 8x + 2) - (2x^2 - 6x) = -2x + 2$

Divide $-2x$ by $x$: get $-2$

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

Subtract:
$(-2x + 2) - (-2x + 6) = -4$

✔ Answer: $2x - 2 - \frac{4}{x - 3}$

---

10.) $(2x^2 + 5x - 11) \div (x - 4)$

Long division:

Divide $2x^2$ by $x$: get $2x$

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

Subtract:
$(2x^2 + 5x - 11) - (2x^2 - 8x) = 13x - 11$

Divide $13x$ by $x$: get $13$

Multiply: $13(x - 4) = 13x - 52$

Subtract:
$(13x - 11) - (13x - 52) = 41$

✔ Answer: $2x + 13 + \frac{41}{x - 4}$

---

✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $3x^2 + 3x + \frac{3}{8}$ |
| 2 | $4x^4 + \frac{1}{2}x^3 + \frac{1}{4}x^2$ |
| 3 | $x^2 + 3x + 1$ |
| 4 | $3x^2 + \frac{1}{2}x - 2$ |
| 5 | $x + 13 + \frac{69}{x - 4}$ |
| 6 | $x + 5 + \frac{4}{x - 4}$ |
| 7 | $x - 5 - \frac{3}{x - 1}$ |
| 8 | $x + 5 + \frac{1}{x - 5}$ |
| 9 | $2x - 2 - \frac{4}{x - 3}$ |
| 10 | $2x + 13 + \frac{41}{x - 4}$ |

Let me know if you'd like these checked or explained further!