Polynomials Worksheets with Answer Key - Free Printable

Let's solve each problem step by step from the Polynomials Practice Worksheet.

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Add the given expressions

#### 1. $(6 + 2n^3) + (5n^2 + 2)$

Group like terms:
- $2n^3$: only one term
- $5n^2$: only one term
- Constants: $6 + 2 = 8$

So,
$$
2n^3 + 5n^2 + 8
$$

✔ Answer: $2n^3 + 5n^2 + 8$

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#### 2. $(6m^5 + 1) + (4m^5 + 9m - 1)$

Combine like terms:
- $6m^5 + 4m^5 = 10m^5$
- $9m$: only one term
- Constants: $1 - 1 = 0$

So,
$$
10m^5 + 9m
$$

✔ Answer: $10m^5 + 9m$

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#### 3. $(3y^2 - y) + 5y^3 + (-4y^3 + y^2 - 8)$

First, rewrite all terms:
$$
3y^2 - y + 5y^3 - 4y^3 + y^2 - 8
$$

Now combine like terms:
- $5y^3 - 4y^3 = y^3$
- $3y^2 + y^2 = 4y^2$
- $-y$: only one
- $-8$: constant

So,
$$
y^3 + 4y^2 - y - 8
$$

✔ Answer: $y^3 + 4y^2 - y - 8$

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#### 4. $(4v^4 - v^2) + (5v^2 + v^4)$

Rewrite:
$$
4v^4 - v^2 + 5v^2 + v^4
$$

Combine:
- $4v^4 + v^4 = 5v^4$
- $-v^2 + 5v^2 = 4v^2$

So,
$$
5v^4 + 4v^2
$$

✔ Answer: $5v^4 + 4v^2$

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Subtract the given expressions

#### 5. $(-2x^3 + x) - (7x - 3 + 7x^3)$

Distribute the negative sign:
$$
-2x^3 + x - 7x + 3 - 7x^3
$$

Now combine:
- $-2x^3 - 7x^3 = -9x^3$
- $x - 7x = -6x$
- $+3$

So,
$$
-9x^3 - 6x + 3
$$

✔ Answer: $-9x^3 - 6x + 3$

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#### 6. $(3a^2 + 2a - 2) - (a^2 - 3a + 7)$

Distribute the negative:
$$
3a^2 + 2a - 2 - a^2 + 3a - 7
$$

Combine:
- $3a^2 - a^2 = 2a^2$
- $2a + 3a = 5a$
- $-2 - 7 = -9$

So,
$$
2a^2 + 5a - 9
$$

✔ Answer: $2a^2 + 5a - 9$

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Multiply the given expressions

#### 7. $(4v - 2)(7v^2 + 6v - 8)$

Use distributive property (FOIL-like for trinomials):

Multiply each term in first polynomial by each in second:

- $4v \cdot 7v^2 = 28v^3$
- $4v \cdot 6v = 24v^2$
- $4v \cdot (-8) = -32v$
- $-2 \cdot 7v^2 = -14v^2$
- $-2 \cdot 6v = -12v$
- $-2 \cdot (-8) = 16$

Now add all:
$$
28v^3 + 24v^2 - 32v - 14v^2 - 12v + 16
$$

Combine like terms:
- $28v^3$
- $24v^2 - 14v^2 = 10v^2$
- $-32v - 12v = -44v$
- $+16$

So,
$$
28v^3 + 10v^2 - 44v + 16
$$

✔ Answer: $28v^3 + 10v^2 - 44v + 16$

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#### 8. $(4x^2 + 3x + 1)(7x^2 - 4x - 2)$

Multiply each term:

Break it down:

Step 1: $4x^2(7x^2 - 4x - 2) = 28x^4 - 16x^3 - 8x^2$

Step 2: $3x(7x^2 - 4x - 2) = 21x^3 - 12x^2 - 6x$

Step 3: $1(7x^2 - 4x - 2) = 7x^2 - 4x - 2$

Now add all together:
$$
28x^4 - 16x^3 - 8x^2 \\
+ 21x^3 - 12x^2 - 6x \\
+ 7x^2 - 4x - 2
$$

Combine like terms:

- $28x^4$
- $-16x^3 + 21x^3 = 5x^3$
- $-8x^2 - 12x^2 + 7x^2 = -13x^2$
- $-6x - 4x = -10x$
- $-2$

So,
$$
28x^4 + 5x^3 - 13x^2 - 10x - 2
$$

✔ Answer: $28x^4 + 5x^3 - 13x^2 - 10x - 2$

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Divide the given expressions

#### 9. $(n^2 - n - 80) \div (n - 9)$

We'll use polynomial long division.

Divide $n^2 - n - 80$ by $n - 9$

1. Divide $n^2 \div n = n$
2. Multiply: $n(n - 9) = n^2 - 9n$
3. Subtract: $(n^2 - n - 80) - (n^2 - 9n) = (-n + 9n) - 80 = 8n - 80$
4. Divide $8n \div n = 8$
5. Multiply: $8(n - 9) = 8n - 72$
6. Subtract: $(8n - 80) - (8n - 72) = -8$

So,
$$
n + 8 \quad \text{with remainder } -8
$$

Thus,
$$
\frac{n^2 - n - 80}{n - 9} = n + 8 - \frac{8}{n - 9}
$$

But since the question likely wants simplified form (possibly factoring), let’s check if the quadratic factors.

Factor $n^2 - n - 80$: look for two numbers that multiply to $-80$, add to $-1$

Try: $-10$ and $8$: $-10 + 8 = -2$ ✘
$-8$ and $10$: $-8 + 10 = 2$ ✘
$-16$ and $5$: $-16 + 5 = -11$ ✘
$-5$ and $16$: $-5 + 16 = 11$ ✘
Wait — $-10$ and $8$: no.

Actually, try:
- $-10$ and $8$: product $-80$, sum $-2$
- $-8$ and $10$: sum $2$
- $-16$ and $5$: sum $-11$
- $-5$ and $16$: sum $11$
- $-20$ and $4$: sum $-16$
- $-4$ and $20$: sum $16$

No integer pair adds to $-1$. So doesn't factor nicely.

But we already did division:
$$
n + 8 - \frac{8}{n - 9}
$$

However, double-check the division:

Wait! Let’s plug in $n = 9$ into numerator: $9^2 - 9 - 80 = 81 - 9 - 80 = -8$

So remainder is $-8$, correct.

But maybe there's a mistake? Wait — perhaps it does factor?

Try factoring: $n^2 - n - 80$

Discriminant: $1 + 320 = 321$, not perfect square → doesn’t factor over integers.

So answer is:
$$
n + 8 - \frac{8}{n - 9}
$$

But let’s recheck the division:

- $n^2 - n - 80$
- Divide by $n - 9$

Step-by-step:

```
n + 8
_________
n - 9 | n² - n - 80
-(n² - 9n)
-----------
8n - 80
-(8n - 72)
---------
-8
```

Yes, correct.

✔ Answer: $n + 8 - \frac{8}{n - 9}$

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#### 10. $(2x^{11} - 5x^7 - 10x^6) \div 2x^3$

Divide each term by $2x^3$:

- $2x^{11} \div 2x^3 = x^{8}$
- $-5x^7 \div 2x^3 = -\frac{5}{2}x^4$
- $-10x^6 \div 2x^3 = -5x^3$

So,
$$
x^8 - \frac{5}{2}x^4 - 5x^3
$$

✔ Answer: $x^8 - \frac{5}{2}x^4 - 5x^3$

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $2n^3 + 5n^2 + 8$ |
| 2 | $10m^5 + 9m$ |
| 3 | $y^3 + 4y^2 - y - 8$ |
| 4 | $5v^4 + 4v^2$ |
| 5 | $-9x^3 - 6x + 3$ |
| 6 | $2a^2 + 5a - 9$ |
| 7 | $28v^3 + 10v^2 - 44v + 16$ |
| 8 | $28x^4 + 5x^3 - 13x^2 - 10x - 2$ |
| 9 | $n + 8 - \frac{8}{n - 9}$ |
|10 | $x^8 - \frac{5}{2}x^4 - 5x^3$ |

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