Adding and Subtracting Polynomials Worksheets with Answer Key - Free Printable

Let's solve each problem step by step, starting with adding and then subtracting polynomials.

---

Adding Polynomials

#### 1. $(6p^2 - q - 4) + (2q^2 + 5q - 5)$

We combine like terms:

- $6p^2$: no other $p^2$ term → stays
- $-q + 5q = 4q$
- $2q^2$: only one $q^2$ term → stays
- $-4 - 5 = -9$

So:
$$
6p^2 + 2q^2 + 4q - 9
$$

✔ Answer: $6p^2 + 2q^2 + 4q - 9$

---

#### 2. $(4x^2 + 2x - 5) + (6x^2 - x - 5)$

Combine like terms:

- $4x^2 + 6x^2 = 10x^2$
- $2x - x = x$
- $-5 - 5 = -10$

So:
$$
10x^2 + x - 10
$$

✔ Answer: $10x^2 + x - 10$

---

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

- $3x^2 + 2x^2 = 5x^2$
- $1 + 4 = 5$

So:
$$
5x^2 + 5
$$

✔ Answer: $5x^2 + 5$

---

#### 4. $3x^3 - 2 + (9x + x^3) + 7$

First, expand the parentheses:
$$
3x^3 - 2 + 9x + x^3 + 7
$$

Now combine like terms:

- $3x^3 + x^3 = 4x^3$
- $9x$: only one → stays
- $-2 + 7 = 5$

So:
$$
4x^3 + 9x + 5
$$

✔ Answer: $4x^3 + 9x + 5$

---

Subtracting Polynomials

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

Distribute the negative sign to all terms in the second polynomial:

$$
4p^2 - p - 3 - p^2 - 2p + 5
$$

Now combine like terms:

- $4p^2 - p^2 = 3p^2$
- $-p - 2p = -3p$
- $-3 + 5 = 2$

So:
$$
3p^2 - 3p + 2
$$

✔ Answer: $3p^2 - 3p + 2$

---

#### 6. $3(c - 4) - 5(c^2 + 4c - 6)$

First, distribute:

- $3(c - 4) = 3c - 12$
- $-5(c^2 + 4c - 6) = -5c^2 - 20c + 30$

Now combine:
$$
3c - 12 - 5c^2 - 20c + 30
$$

Group like terms:

- $-5c^2$
- $3c - 20c = -17c$
- $-12 + 30 = 18$

So:
$$
-5c^2 - 17c + 18
$$

✔ Answer: $-5c^2 - 17c + 18$

---

#### 7. $2(l - 7) - 3(l^2 - 2l + 8)$

Distribute:

- $2(l - 7) = 2l - 14$
- $-3(l^2 - 2l + 8) = -3l^2 + 6l - 24$

Now combine:
$$
2l - 14 - 3l^2 + 6l - 24
$$

Group like terms:

- $-3l^2$
- $2l + 6l = 8l$
- $-14 - 24 = -38$

So:
$$
-3l^2 + 8l - 38
$$

✔ Answer: $-3l^2 + 8l - 38$

---

#### 8. $5(q + 3) - 9(q^2 - 3q + 2)$

Distribute:

- $5(q + 3) = 5q + 15$
- $-9(q^2 - 3q + 2) = -9q^2 + 27q - 18$

Now combine:
$$
5q + 15 - 9q^2 + 27q - 18
$$

Group like terms:

- $-9q^2$
- $5q + 27q = 32q$
- $15 - 18 = -3$

So:
$$
-9q^2 + 32q - 3
$$

✔ Answer: $-9q^2 + 32q - 3$

---

✔ Final Answers Summary:

1. $6p^2 + 2q^2 + 4q - 9$
2. $10x^2 + x - 10$
3. $5x^2 + 5$
4. $4x^3 + 9x + 5$
5. $3p^2 - 3p + 2$
6. $-5c^2 - 17c + 18$
7. $-3l^2 + 8l - 38$
8. $-9q^2 + 32q - 3$

Let me know if you'd like this formatted as a printable worksheet or need explanations for any step!