Simplifying Algebraic Expression Worksheets - Free Printable

It looks like you've uploaded a worksheet titled "Simplifying Polynomial Expressions" with several problems involving simplifying algebraic expressions. Since I can't view images directly, I’ll need you to describe the content of the image or type out the problems so I can help solve them.

However, based on your description and the visible text in your message, here are the problems listed:

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Simplify each expression:

1. $ 3(8b - 2) - (5b + 4) - 6(2b - 6) $

2. $ \frac{2}{3}(x^2 + 3x)^2 - 3(x^2 + 3x) + 3(x^2 + 3x)^2 $

3. $ (2m^2 + 3m)(2m^2 + 3m)^2 $

4. $ 10x^2 + 2(3x^2 - 4x) - 4(3x^2 + 6) $

5. $ 10(2b^2 - 4b^2 + 3a^2) - 2(5b^2 + 3a^2) + a $

6. $ \frac{2}{5}m^2 + 4m^2 - 2m^2 + \frac{10}{7}m^2 + 10m $

7. $ 2(12d^2 + 14d^2 + 10d) - 2(6d^2 + 3d^2 + 8d^2) $

8. $ 3y - 4(y^2 - 3y^2 + 3y^2) $

9. $ \frac{1}{2}(x^2 + 2x^2 + 4x^2 + 4x^2 - 10x) $

10. $ 10p^2 + 2(3p^2 - 6p^2) + 2(7p^2) $

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Let me go through each problem step by step and simplify them.

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Problem 1:

$ 3(8b - 2) - (5b + 4) - 6(2b - 6) $

Step 1: Distribute:
- $ 3(8b - 2) = 24b - 6 $
- $ -(5b + 4) = -5b - 4 $
- $ -6(2b - 6) = -12b + 36 $

Step 2: Combine all terms:
$$
24b - 6 - 5b - 4 - 12b + 36
$$

Step 3: Combine like terms:
- $ (24b - 5b - 12b) = 7b $
- $ (-6 - 4 + 36) = 26 $

✔ Answer: $ 7b + 26 $

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Problem 2:

$ \frac{2}{3}(x^2 + 3x)^2 - 3(x^2 + 3x) + 3(x^2 + 3x)^2 $

Let $ u = x^2 + 3x $. Then the expression becomes:
$$
\frac{2}{3}u^2 - 3u + 3u^2
$$

Combine like terms:
- $ \frac{2}{3}u^2 + 3u^2 = \left(\frac{2}{3} + \frac{9}{3}\right)u^2 = \frac{11}{3}u^2 $
- So: $ \frac{11}{3}u^2 - 3u $

Now substitute back $ u = x^2 + 3x $:
$$
\frac{11}{3}(x^2 + 3x)^2 - 3(x^2 + 3x)
$$

We could expand further if needed, but unless specified, this is simplified enough.

✔ Answer: $ \frac{11}{3}(x^2 + 3x)^2 - 3(x^2 + 3x) $

> Optional expansion:
- $ (x^2 + 3x)^2 = x^4 + 6x^3 + 9x^2 $
- So $ \frac{11}{3}(x^4 + 6x^3 + 9x^2) = \frac{11}{3}x^4 + 22x^3 + 33x^2 $
- $ -3(x^2 + 3x) = -3x^2 - 9x $
- Final: $ \frac{11}{3}x^4 + 22x^3 + 30x^2 - 9x $

But unless asked, keep it factored.

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Problem 3:

$ (2m^2 + 3m)(2m^2 + 3m)^2 $

Note: This is $ (2m^2 + 3m)^1 \cdot (2m^2 + 3m)^2 = (2m^2 + 3m)^3 $

So:
$$
(2m^2 + 3m)^3
$$

You can factor $ m $ from the binomial:
$ 2m^2 + 3m = m(2m + 3) $

So:
$$
[m(2m + 3)]^3 = m^3(2m + 3)^3
$$

✔ Answer: $ m^3(2m + 3)^3 $

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Problem 4:

$ 10x^2 + 2(3x^2 - 4x) - 4(3x^2 + 6) $

Distribute:
- $ 2(3x^2 - 4x) = 6x^2 - 8x $
- $ -4(3x^2 + 6) = -12x^2 - 24 $

Now combine:
$$
10x^2 + 6x^2 - 8x - 12x^2 - 24
$$

Combine like terms:
- $ (10 + 6 - 12)x^2 = 4x^2 $
- $ -8x $
- $ -24 $

✔ Answer: $ 4x^2 - 8x - 24 $

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Problem 5:

$ 10(2b^2 - 4b^2 + 3a^2) - 2(5b^2 + 3a^2) + a $

First simplify inside parentheses:
- $ 2b^2 - 4b^2 = -2b^2 $
- So: $ 10(-2b^2 + 3a^2) = -20b^2 + 30a^2 $
- $ -2(5b^2 + 3a^2) = -10b^2 - 6a^2 $
- Add $ +a $

Now combine:
- $ -20b^2 - 10b^2 = -30b^2 $
- $ 30a^2 - 6a^2 = 24a^2 $
- $ +a $

✔ Answer: $ -30b^2 + 24a^2 + a $

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Problem 6:

$ \frac{2}{5}m^2 + 4m^2 - 2m^2 + \frac{10}{7}m^2 + 10m $

Combine $ m^2 $ terms:
- $ \frac{2}{5} + 4 - 2 + \frac{10}{7} $
- Convert to common denominator (LCM of 5 and 7 is 35):

- $ \frac{2}{5} = \frac{14}{35} $
- $ 4 = \frac{140}{35} $
- $ -2 = -\frac{70}{35} $
- $ \frac{10}{7} = \frac{50}{35} $

Sum: $ \frac{14 + 140 - 70 + 50}{35} = \frac{134}{35} $

So: $ \frac{134}{35}m^2 + 10m $

✔ Answer: $ \frac{134}{35}m^2 + 10m $

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Problem 7:

$ 2(12d^2 + 14d^2 + 10d) - 2(6d^2 + 3d^2 + 8d^2) $

First simplify inside:
- $ 12d^2 + 14d^2 = 26d^2 $ → $ 2(26d^2 + 10d) = 52d^2 + 20d $
- $ 6d^2 + 3d^2 + 8d^2 = 17d^2 $ → $ -2(17d^2) = -34d^2 $

Now combine:
- $ 52d^2 - 34d^2 = 18d^2 $
- $ +20d $

✔ Answer: $ 18d^2 + 20d $

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Problem 8:

$ 3y - 4(y^2 - 3y^2 + 3y^2) $

Simplify inside parentheses:
- $ y^2 - 3y^2 + 3y^2 = (1 - 3 + 3)y^2 = 1y^2 $
- So: $ 3y - 4(y^2) = 3y - 4y^2 $

✔ Answer: $ -4y^2 + 3y $

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Problem 9:

$ \frac{1}{2}(x^2 + 2x^2 + 4x^2 + 4x^2 - 10x) $

Combine like terms inside:
- $ x^2 + 2x^2 + 4x^2 + 4x^2 = 11x^2 $
- So: $ \frac{1}{2}(11x^2 - 10x) = \frac{11}{2}x^2 - 5x $

✔ Answer: $ \frac{11}{2}x^2 - 5x $

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Problem 10:

$ 10p^2 + 2(3p^2 - 6p^2) + 2(7p^2) $

Simplify:
- $ 3p^2 - 6p^2 = -3p^2 $
- $ 2(-3p^2) = -6p^2 $
- $ 2(7p^2) = 14p^2 $

Now combine:
- $ 10p^2 - 6p^2 + 14p^2 = 18p^2 $

✔ Answer: $ 18p^2 $

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✔ Final Answers Summary:

1. $ 7b + 26 $
2. $ \frac{11}{3}(x^2 + 3x)^2 - 3(x^2 + 3x) $
3. $ m^3(2m + 3)^3 $
4. $ 4x^2 - 8x - 24 $
5. $ -30b^2 + 24a^2 + a $
6. $ \frac{134}{35}m^2 + 10m $
7. $ 18d^2 + 20d $
8. $ -4y^2 + 3y $
9. $ \frac{11}{2}x^2 - 5x $
10. $ 18p^2 $

Let me know if you'd like any of these expanded further or explained in more detail!