Long Division Worksheets| Download Free Printables For Kids - Free Printable

Let's solve each of the polynomial long division problems step by step. We'll go through each one carefully using the long division method for polynomials, which is similar to numerical long division.

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🔹 Problem 1:

$$
(2x^2 - 5x + 3) \div (2x - 1)
$$

#### Step-by-step:

We divide $ 2x^2 - 5x + 3 $ by $ 2x - 1 $.

1. Divide leading terms:
$$
\frac{2x^2}{2x} = x
$$
Write $ x $ above the division bar.

2. Multiply $ x \cdot (2x - 1) = 2x^2 - x $

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

4. Bring down (already there): $ -4x + 3 $

5. Divide: $ \frac{-4x}{2x} = -2 $

6. Multiply: $ -2 \cdot (2x - 1) = -4x + 2 $

7. Subtract:
$$
(-4x + 3) - (-4x + 2) = 3 - 2 = 1
$$

✔ Quotient: $ x - 2 $, Remainder: $ 1 $

So:
$$
\frac{2x^2 - 5x + 3}{2x - 1} = x - 2 + \frac{1}{2x - 1}
$$

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🔹 Problem 2:

$$
(m^2 - 6m + 1) \div (m - 4)
$$

1. Divide $ m^2 \div m = m $

2. Multiply: $ m(m - 4) = m^2 - 4m $

3. Subtract:
$$
(m^2 - 6m + 1) - (m^2 - 4m) = -6m + 4m + 1 = -2m + 1
$$

4. Divide: $ -2m \div m = -2 $

5. Multiply: $ -2(m - 4) = -2m + 8 $

6. Subtract:
$$
(-2m + 1) - (-2m + 8) = 1 - 8 = -7
$$

✔ Quotient: $ m - 2 $, Remainder: $ -7 $

So:
$$
\frac{m^2 - 6m + 1}{m - 4} = m - 2 - \frac{7}{m - 4}
$$

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🔹 Problem 3:

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

Note: $ x^2 - 9 = (x - 3)(x + 3) $ → This is a difference of squares

So:
$$
\frac{x^2 - 9}{x + 3} = x - 3
$$

✔ Quotient: $ x - 3 $, Remainder: $ 0 $

But let’s do it via long division:

1. $ x^2 \div x = x $

2. $ x(x + 3) = x^2 + 3x $

3. Subtract: $ (x^2 - 9) - (x^2 + 3x) = -3x - 9 $

4. $ -3x \div x = -3 $

5. $ -3(x + 3) = -3x - 9 $

6. Subtract: $ (-3x - 9) - (-3x - 9) = 0 $

✔ Quotient: $ x - 3 $, Remainder: $ 0 $

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🔹 Problem 4:

$$
(x^2 + 2 - 5x) \div (x - 3)
$$

First, write in standard form: $ x^2 - 5x + 2 $

Now divide $ x^2 - 5x + 2 $ by $ x - 3 $

1. $ x^2 \div x = x $

2. $ x(x - 3) = x^2 - 3x $

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

4. $ -2x \div x = -2 $

5. $ -2(x - 3) = -2x + 6 $

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

✔ Quotient: $ x - 2 $, Remainder: $ -4 $

So:
$$
\frac{x^2 - 5x + 2}{x - 3} = x - 2 - \frac{4}{x - 3}
$$

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🔹 Problem 5:

$$
(x^2 + 4) \div (x + 1)
$$

Divide $ x^2 + 0x + 4 $ by $ x + 1 $

1. $ x^2 \div x = x $

2. $ x(x + 1) = x^2 + x $

3. Subtract: $ (x^2 + 0x + 4) - (x^2 + x) = -x + 4 $

4. $ -x \div x = -1 $

5. $ -1(x + 1) = -x - 1 $

6. Subtract: $ (-x + 4) - (-x - 1) = 4 + 1 = 5 $

✔ Quotient: $ x - 1 $, Remainder: $ 5 $

So:
$$
\frac{x^2 + 4}{x + 1} = x - 1 + \frac{5}{x + 1}
$$

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🔹 Problem 6:

$$
(5y^2 - 6y + 7) \div (5y - 1)
$$

1. $ 5y^2 \div 5y = y $

2. $ y(5y - 1) = 5y^2 - y $

3. Subtract: $ (5y^2 - 6y + 7) - (5y^2 - y) = -6y + y + 7 = -5y + 7 $

4. $ -5y \div 5y = -1 $

5. $ -1(5y - 1) = -5y + 1 $

6. Subtract: $ (-5y + 7) - (-5y + 1) = 7 - 1 = 6 $

✔ Quotient: $ y - 1 $, Remainder: $ 6 $

So:
$$
\frac{5y^2 - 6y + 7}{5y - 1} = y - 1 + \frac{6}{5y - 1}
$$

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✔ Final Answers Summary:

| Problem | Quotient | Remainder | Result |
|--------|----------|-----------|--------|
| 1 | $ x - 2 $ | $ 1 $ | $ x - 2 + \frac{1}{2x - 1} $ |
| 2 | $ m - 2 $ | $ -7 $ | $ m - 2 - \frac{7}{m - 4} $ |
| 3 | $ x - 3 $ | $ 0 $ | $ x - 3 $ |
| 4 | $ x - 2 $ | $ -4 $ | $ x - 2 - \frac{4}{x - 3} $ |
| 5 | $ x - 1 $ | $ 5 $ | $ x - 1 + \frac{5}{x + 1} $ |
| 6 | $ y - 1 $ | $ 6 $ | $ y - 1 + \frac{6}{5y - 1} $ |

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