Worksheet: Dividing Polynomials - Long & Synthetic Division ... - Free Printable

It looks like you've uploaded a worksheet titled "Division of Algebraic Expressions" with six problems involving simplifying algebraic fractions. Since I can't view images directly, I'll assume the problems are as written in your text and solve them step by step.

Here are the problems based on your description:

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(1)

$$
\frac{6a^2b}{3ab}
$$

Step-by-step:

- Cancel common factors in numerator and denominator.
- $ \frac{6a^2b}{3ab} = \frac{6}{3} \cdot \frac{a^2}{a} \cdot \frac{b}{b} $
- $ = 2 \cdot a^{2-1} \cdot 1 = 2a $

✔ Answer: $ \boxed{2a} $

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(2)

$$
\frac{10x^2y - 15xy^2}{5xy}
$$

Step-by-step:

- Split into two separate fractions:
$$
\frac{10x^2y}{5xy} - \frac{15xy^2}{5xy}
$$

- Simplify each term:
- $ \frac{10x^2y}{5xy} = \frac{10}{5} \cdot \frac{x^2}{x} \cdot \frac{y}{y} = 2x $
- $ \frac{15xy^2}{5xy} = \frac{15}{5} \cdot \frac{x}{x} \cdot \frac{y^2}{y} = 3y $

So,
$$
2x - 3y
$$

✔ Answer: $ \boxed{2x - 3y} $

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(3)

$$
\frac{8a^2 - 16a^2}{4a}
$$

Wait — this seems odd: $ 8a^2 - 16a^2 = -8a^2 $. Let's check:

$$
\frac{8a^2 - 16a^2}{4a} = \frac{-8a^2}{4a}
$$

Now simplify:
- $ \frac{-8a^2}{4a} = \frac{-8}{4} \cdot \frac{a^2}{a} = -2a $

✔ Answer: $ \boxed{-2a} $

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(4)

$$
\frac{-7m^2n^3 + 28m^2n^2}{-4mn^2}
$$

Step-by-step:

Split into two terms:
$$
\frac{-7m^2n^3}{-4mn^2} + \frac{28m^2n^2}{-4mn^2}
$$

Simplify each:

1. $ \frac{-7m^2n^3}{-4mn^2} = \frac{7}{4} \cdot \frac{m^2}{m} \cdot \frac{n^3}{n^2} = \frac{7}{4} m n $

2. $ \frac{28m^2n^2}{-4mn^2} = -7 \cdot \frac{m^2}{m} \cdot \frac{n^2}{n^2} = -7m $

So:
$$
\frac{7}{4}mn - 7m
$$

Factor out $ m $:
$$
m\left(\frac{7}{4}n - 7\right)
$$

But unless asked to factor, we can leave it as:
$$
\frac{7}{4}mn - 7m
$$

✔ Answer: $ \boxed{\frac{7}{4}mn - 7m} $

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(5)

$$
\frac{2a^2b - 4a^2c + 2a^2d}{2a}
$$

Step-by-step:

Factor numerator first:
- All terms have $ 2a^2 $: $ 2a^2(b - 2c + d) $

So:
$$
\frac{2a^2(b - 2c + d)}{2a} = \frac{2a^2}{2a} \cdot (b - 2c + d) = a(b - 2c + d)
$$

✔ Answer: $ \boxed{a(b - 2c + d)} $

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(6)

$$
\frac{9x^2y^2 - 3x^2y + 9xy^2 - 3xy}{3xy}
$$

Step-by-step:

Split into four terms:
$$
\frac{9x^2y^2}{3xy} - \frac{3x^2y}{3xy} + \frac{9xy^2}{3xy} - \frac{3xy}{3xy}
$$

Simplify each:

1. $ \frac{9x^2y^2}{3xy} = 3xy $
2. $ \frac{3x^2y}{3xy} = x $
3. $ \frac{9xy^2}{3xy} = 3y $
4. $ \frac{3xy}{3xy} = 1 $

So:
$$
3xy - x + 3y - 1
$$

✔ Answer: $ \boxed{3xy - x + 3y - 1} $

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✔ Final Answers:

1. $ 2a $
2. $ 2x - 3y $
3. $ -2a $
4. $ \frac{7}{4}mn - 7m $
5. $ a(b - 2c + d) $
6. $ 3xy - x + 3y - 1 $

Let me know if you'd like these explained differently or want to check for factoring alternatives!