Simplifying Polynomials worksheets - Free Printable

Sure! Let’s go through each problem one by one and simplify the polynomial expressions. The goal is to combine like terms, distribute where needed, and write the expression in its simplest form.

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1) \( 3x^2 - 5x^3 - x(2x^2 + 4x) \)

First, distribute the \(-x\):

\[
-x(2x^2 + 4x) = -2x^3 - 4x^2
\]

Now substitute back:

\[
3x^2 - 5x^3 - 2x^3 - 4x^2
\]

Combine like terms:

- \( -5x^3 - 2x^3 = -7x^3 \)
- \( 3x^2 - 4x^2 = -x^2 \)

✔ Answer: \( -7x^3 - x^2 \)

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2) \( 6(8r - 5) \)

Distribute the 6:

\[
6 \cdot 8r = 48r,\quad 6 \cdot (-5) = -30
\]

✔ Answer: \( 48r - 30 \)

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3) \( (2v + 6)(4v) \)

Distribute \(4v\):

\[
4v \cdot 2v = 8v^2,\quad 4v \cdot 6 = 24v
\]

✔ Answer: \( 8v^2 + 24v \)

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4) \( a^2 - 2a + 5a^3 + 1 - 10a \)

Group like terms:

- Cubic: \( 5a^3 \)
- Quadratic: \( a^2 \)
- Linear: \( -2a - 10a = -12a \)
- Constant: \( +1 \)

✔ Answer: \( 5a^3 + a^2 - 12a + 1 \)

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5) \( x(-2x + 5x^2) \)

Distribute \(x\):

\[
x \cdot (-2x) = -2x^2,\quad x \cdot 5x^2 = 5x^3
\]

✔ Answer: \( 5x^3 - 2x^2 \)

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6) \( 2x^3 - 4x^3 + 2x^4 + 1 \)

Combine like terms:

- \( 2x^3 - 4x^3 = -2x^3 \)
- \( 2x^4 \) stays
- \( +1 \) stays

✔ Answer: \( 2x^4 - 2x^3 + 1 \)

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7) \( 3y^2(1 - y - 2y^2) \)

Distribute \(3y^2\):

\[
3y^2 \cdot 1 = 3y^2 \\
3y^2 \cdot (-y) = -3y^3 \\
3y^2 \cdot (-2y^2) = -6y^4
\]

✔ Answer: \( -6y^4 - 3y^3 + 3y^2 \)

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8) \( 10x^2 - 4 - 2x^2 + x^2 \)

Combine like terms:

- \( 10x^2 - 2x^2 + x^2 = 9x^2 \)
- Constant: \( -4 \)

✔ Answer: \( 9x^2 - 4 \)

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9) \( -5g(3 - 2g^2) \)

Distribute \(-5g\):

\[
-5g \cdot 3 = -15g \\
-5g \cdot (-2g^2) = +10g^3
\]

✔ Answer: \( 10g^3 - 15g \)

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10) \( 1 + 3x + 4x^2 - 5x + 8 \)

Combine like terms:

- Constants: \(1 + 8 = 9\)
- Linear: \(3x - 5x = -2x\)
- Quadratic: \(4x^2\)

✔ Answer: \( 4x^2 - 2x + 9 \)

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11) \( 3a + b - 4b + 5a \)

Combine like terms:

- \(3a + 5a = 8a\)
- \(b - 4b = -3b\)

✔ Answer: \( 8a - 3b \)

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12) \( (7n + 8)(-4) \)

Distribute \(-4\):

\[
-4 \cdot 7n = -28n,\quad -4 \cdot 8 = -32
\]

✔ Answer: \( -28n - 32 \)

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13) \( -2x^3 + 8x^3 + 9x^3 + 5 \)

Combine like terms:

- \( -2x^3 + 8x^3 + 9x^3 = 15x^3 \)
- Constant: \( +5 \)

✔ Answer: \( 15x^3 + 5 \)

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14) \( (3x + 4)(3x) \)

Distribute \(3x\):

\[
3x \cdot 3x = 9x^2,\quad 3x \cdot 4 = 12x
\]

✔ Answer: \( 9x^2 + 12x \)

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15) \( 5m^2 - 7m^3 + 3m^2 + 3 \)

Combine like terms:

- Cubic: \( -7m^3 \)
- Quadratic: \(5m^2 + 3m^2 = 8m^2\)
- Constant: \( +3 \)

✔ Answer: \( -7m^3 + 8m^2 + 3 \)

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16) \( 3x(1 + 6x - 5) \)

First simplify inside parentheses:

\[
1 - 5 = -4,\quad so: 6x - 4
\]

Now distribute \(3x\):

\[
3x \cdot 6x = 18x^2,\quad 3x \cdot (-4) = -12x
\]

✔ Answer: \( 18x^2 - 12x \)

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17) \( (2x - 6)(7x) \)

Distribute \(7x\):

\[
7x \cdot 2x = 14x^2,\quad 7x \cdot (-6) = -42x
\]

✔ Answer: \( 14x^2 - 42x \)

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18) \( 12x + 10 - 2x - 8 \)

Combine like terms:

- Linear: \(12x - 2x = 10x\)
- Constants: \(10 - 8 = 2\)

✔ Answer: \( 10x + 2 \)

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19) \( 5x + 10 - 2x + 5 \)

Combine like terms:

- Linear: \(5x - 2x = 3x\)
- Constants: \(10 + 5 = 15\)

✔ Answer: \( 3x + 15 \)

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20) \( 2x(3x + 5) \)

Distribute \(2x\):

\[
2x \cdot 3x = 6x^2,\quad 2x \cdot 5 = 10x
\]

✔ Answer: \( 6x^2 + 10x \)

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## ✔ Final Answers Summary:

1) \( -7x^3 - x^2 \)
2) \( 48r - 30 \)
3) \( 8v^2 + 24v \)
4) \( 5a^3 + a^2 - 12a + 1 \)
5) \( 5x^3 - 2x^2 \)
6) \( 2x^4 - 2x^3 + 1 \)
7) \( -6y^4 - 3y^3 + 3y^2 \)
8) \( 9x^2 - 4 \)
9) \( 10g^3 - 15g \)
10) \( 4x^2 - 2x + 9 \)
11) \( 8a - 3b \)
12) \( -28n - 32 \)
13) \( 15x^3 + 5 \)
14) \( 9x^2 + 12x \)
15) \( -7m^3 + 8m^2 + 3 \)
16) \( 18x^2 - 12x \)
17) \( 14x^2 - 42x \)
18) \( 10x + 2 \)
19) \( 3x + 15 \)
20) \( 6x^2 + 10x \)

Let me know if you’d like these written out in a printable format or explained further!