Simplifying Polynomials worksheets - Free Printable

Let's solve each of the 20 polynomial simplification problems step by step.

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

First, distribute $-x$ into the parentheses:
$$
-x(2x^2 + 4x) = -2x^3 - 4x^2
$$
Now combine all terms:
$$
3x^2 - 5x^3 - 2x^3 - 4x^2 = (-5x^3 - 2x^3) + (3x^2 - 4x^2) = -7x^3 - x^2
$$
✔ Answer: $-7x^3 - x^2$

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2) $6(8r - 5)$

Distribute 6:
$$
6 \cdot 8r = 48r,\quad 6 \cdot (-5) = -30
$$
✔ Answer: $48r - 30$

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

Distribute $4v$:
$$
2v \cdot 4v = 8v^2,\quad 6 \cdot 4v = 24v
$$
✔ Answer: $8v^2 + 24v$

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4) $a^2 - 2a + 5a^3 + 1 - 10a$

Combine like terms:
- $a^2$: only one term
- $-2a - 10a = -12a$
- $5a^3$: only one
- $+1$: constant

Arrange in descending order:
$$
5a^3 + a^2 - 12a + 1
$$
✔ Answer: $5a^3 + a^2 - 12a + 1$

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

Distribute $x$:
$$
x \cdot (-2x) = -2x^2,\quad x \cdot 5x^2 = 5x^3
$$
So: $5x^3 - 2x^2$

✔ Answer: $5x^3 - 2x^2$

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6) $2x^3 - 4x^3 + 2x^4 + 1$

Combine like terms:
- $2x^3 - 4x^3 = -2x^3$
- $2x^4$: only one
- $+1$: constant

Order by degree:
$$
2x^4 - 2x^3 + 1
$$
✔ Answer: $2x^4 - 2x^3 + 1$

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

Distribute $3y^2$:
$$
3y^2 \cdot 1 = 3y^2 \\
3y^2 \cdot (-y) = -3y^3 \\
3y^2 \cdot (-2y^2) = -6y^4
$$
So: $-6y^4 - 3y^3 + 3y^2$

✔ Answer: $-6y^4 - 3y^3 + 3y^2$

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8) $10x^2 - 4 - 2x^2 + x^2$

Combine like terms:
- $10x^2 - 2x^2 + x^2 = 9x^2$
- $-4$: constant

✔ Answer: $9x^2 - 4$

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

Distribute $-5g$:
$$
-5g \cdot 3 = -15g \\
-5g \cdot (-2g^2) = +10g^3
$$
So: $10g^3 - 15g$

✔ Answer: $10g^3 - 15g$

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

Combine like terms:
- Constants: $1 + 8 = 9$
- $3x - 5x = -2x$
- $4x^2$: only one

Order: $4x^2 - 2x + 9$

✔ 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 $-4$:
$$
-4 \cdot 7n = -28n \\
-4 \cdot 8 = -32
$$
✔ Answer: $-28n - 32$

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13) $-2x^3 + 8x^3 + 9x^3 + 5$

Combine like terms:
- $-2 + 8 + 9 = 15$, so $15x^3$
- $+5$: constant

✔ Answer: $15x^3 + 5$

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14) $(3x + 4)(3x)$

Distribute $3x$:
$$
3x \cdot 3x = 9x^2 \\
3x \cdot 4 = 12x
$$
✔ Answer: $9x^2 + 12x$

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15) $5m^2 - 7m^3 + 3m^2 + 3$

Combine like terms:
- $5m^2 + 3m^2 = 8m^2$
- $-7m^3$: only one
- $+3$: constant

Order: $-7m^3 + 8m^2 + 3$

✔ Answer: $-7m^3 + 8m^2 + 3$

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

Simplify inside first: $1 - 5 = -4$, so:
$$
3x(6x - 4)
$$
Now distribute:
$$
3x \cdot 6x = 18x^2 \\
3x \cdot (-4) = -12x
$$
✔ Answer: $18x^2 - 12x$

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

Distribute $7x$:
$$
2x \cdot 7x = 14x^2 \\
-6 \cdot 7x = -42x
$$
✔ Answer: $14x^2 - 42x$

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18) $12x + 10 - 2x - 8$

Combine like terms:
- $12x - 2x = 10x$
- $10 - 8 = 2$

✔ Answer: $10x + 2$

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19) $5x + 10 - 2x + 5$

Combine:
- $5x - 2x = 3x$
- $10 + 5 = 15$

✔ Answer: $3x + 15$

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

Distribute $2x$:
$$
2x \cdot 3x = 6x^2 \\
2x \cdot 5 = 10x
$$
✔ Answer: $6x^2 + 10x$

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## ✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 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 | $6x^2 + 10x$ |

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