Polynomial operations practice worksheet with algebraic expressions.

Let’s solve each problem step by step. We’ll start with the “Adding, Subtracting & Multiplying Polynomials” section — specifically, simplifying expressions.

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Problem 1:
(3x² + 4) – (2x² – 5)

Step 1: Remove parentheses. Remember to distribute the minus sign to both terms inside the second set of parentheses.

→ 3x² + 4 – 2x² + 5

Step 2: Combine like terms.

- x² terms: 3x² – 2x² = x²
- constants: 4 + 5 = 9

Final answer for #1: x² + 9

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Problem 2:
(5a³ – 6a² + 7) – (3a³ – a² – 4)

Step 1: Distribute the minus sign.

→ 5a³ – 6a² + 7 – 3a³ + a² + 4

Step 2: Combine like terms.

- a³: 5a³ – 3a³ = 2a³
- a²: –6a² + a² = –5a²
- constants: 7 + 4 = 11

Final answer for #2: 2a³ – 5a² + 11

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Problem 3:
(8m⁴ – 3m² + m) – (–2m⁴ + 5m² – 6)

Step 1: Distribute the minus sign (which flips all signs in the second polynomial).

→ 8m⁴ – 3m² + m + 2m⁴ – 5m² + 6

Step 2: Combine like terms.

- m⁴: 8m⁴ + 2m⁴ = 10m⁴
- m²: –3m² – 5m² = –8m²
- m term: just +m
- constant: +6

Final answer for #3: 10m⁴ – 8m² + m + 6

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Problem 4:
(9y³ + 2y – 1) – (4y³ – y + 3)

Step 1: Distribute the minus sign.

→ 9y³ + 2y – 1 – 4y³ + y – 3

Step 2: Combine like terms.

- y³: 9y³ – 4y³ = 5y³
- y: 2y + y = 3y
- constants: –1 – 3 = –4

Final answer for #4: 5y³ + 3y – 4

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Now let’s do the multiplication problems (“Multiply Polynomials: Find each product”).

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Problem 5:
7(x + 3)(x – 2)

We can multiply the two binomials first, then multiply by 7.

First: (x + 3)(x – 2)

Use FOIL:

- First: x * x = x²
- Outer: x * (-2) = -2x
- Inner: 3 * x = 3x
- Last: 3 * (-2) = -6

Combine: x² – 2x + 3x – 6 = x² + x – 6

Now multiply by 7:

7(x² + x – 6) = 7x² + 7x – 42

Final answer for #5: 7x² + 7x – 42

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Problem 6:
3(2a – 1)(a + 4)

First multiply (2a – 1)(a + 4):

FOIL:

- First: 2a * a = 2a²
- Outer: 2a * 4 = 8a
- Inner: –1 * a = –a
- Last: –1 * 4 = –4

Combine: 2a² + 8a – a – 4 = 2a² + 7a – 4

Now multiply by 3:

3(2a² + 7a – 4) = 6a² + 21a – 12

Final answer for #6: 6a² + 21a – 12

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Problem 7:
–2b(b – 4)

Distribute –2b to both terms:

–2b * b = –2b²
–2b * (–4) = +8b

Final answer for #7: –2b² + 8b

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Problem 8:
5k(k – 3)(k + 1)

First multiply (k – 3)(k + 1):

FOIL:

- k*k = k²
- k*1 = k
- –3*k = –3k
- –3*1 = –3

Combine: k² + k – 3k – 3 = k² – 2k – 3

Now multiply by 5k:

5k(k² – 2k – 3) = 5k³ – 10k² – 15k

Final answer for #8: 5k³ – 10k² – 15k

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Problem 9:
(3x + 4)(2x – 5)

FOIL:

- 3x * 2x = 6x²
- 3x * (–5) = –15x
- 4 * 2x = 8x
- 4 * (–5) = –20

Combine: 6x² – 15x + 8x – 20 = 6x² – 7x – 20

Final answer for #9: 6x² – 7x – 20

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Problem 10:
(2p – 3)(p + 5)

FOIL:

- 2p * p = 2p²
- 2p * 5 = 10p
- –3 * p = –3p
- –3 * 5 = –15

Combine: 2p² + 10p – 3p – 15 = 2p² + 7p – 15

Final answer for #10: 2p² + 7p – 15

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Problem 11:
(4n – 1)(n + 2)

FOIL:

- 4n * n = 4n²
- 4n * 2 = 8n
- –1 * n = –n
- –1 * 2 = –2

Combine: 4n² + 8n – n – 2 = 4n² + 7n – 2

Final answer for #11: 4n² + 7n – 2

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Problem 12:
(5t + 2)(3t – 4)

FOIL:

- 5t * 3t = 15t²
- 5t * (–4) = –20t
- 2 * 3t = 6t
- 2 * (–4) = –8

Combine: 15t² – 20t + 6t – 8 = 15t² – 14t – 8

Final answer for #12: 15t² – 14t – 8

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Problem 13:
(6w – 5)(w + 3)

FOIL:

- 6w * w = 6w²
- 6w * 3 = 18w
- –5 * w = –5w
- –5 * 3 = –15

Combine: 6w² + 18w – 5w – 15 = 6w² + 13w – 15

Final answer for #13: 6w² + 13w – 15

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Problem 14:
(7z + 1)(z – 2)

FOIL:

- 7z * z = 7z²
- 7z * (–2) = –14z
- 1 * z = z
- 1 * (–2) = –2

Combine: 7z² – 14z + z – 2 = 7z² – 13z – 2

Final answer for #14: 7z² – 13z – 2

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Problem 15:
(2x² + 3x – 1)(x + 4)

This is a trinomial times a binomial. Multiply each term in the trinomial by each term in the binomial.

First, multiply by x:

x*(2x²) = 2x³
x*(3x) = 3x²
x*(–1) = –x

Then multiply by 4:

4*(2x²) = 8x²
4*(3x) = 12x
4*(–1) = –4

Now combine all:

2x³ + 3x² – x + 8x² + 12x – 4

Group like terms:

- x³: 2x³
- x²: 3x² + 8x² = 11x²
- x: –x + 12x = 11x
- constant: –4

Final answer for #15: 2x³ + 11x² + 11x – 4

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Problem 16:
(3y² – 2y + 5)(y – 1)

Multiply each term:

By y:

y*(3y²) = 3y³
y*(–2y) = –2y²
y*(5) = 5y

By –1:

–1*(3y²) = –3y²
–1*(–2y) = +2y
–1*(5) = –5

Combine all:

3y³ – 2y² + 5y – 3y² + 2y – 5

Group like terms:

- y³: 3y³
- y²: –2y² – 3y² = –5y²
- y: 5y + 2y = 7y
- constant: –5

Final answer for #16: 3y³ – 5y² + 7y – 5

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Problem 17:
(4a² + a – 3)(2a + 1)

Multiply:

By 2a:

2a*(4a²) = 8a³
2a*(a) = 2a²
2a*(–3) = –6a

By 1:

1*(4a²) = 4a²
1*(a) = a
1*(–3) = –3

Combine:

8a³ + 2a² – 6a + 4a² + a – 3

Group:

- a³: 8a³
- a²: 2a² + 4a² = 6a²
- a: –6a + a = –5a
- constant: –3

Final answer for #17: 8a³ + 6a² – 5a – 3

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Problem 18:
(5b² – 3b + 2)(b – 4)

Multiply:

By b:

b*(5b²) = 5b³
b*(–3b) = –3b²
b*(2) = 2b

By –4:

–4*(5b²) = –20b²
–4*(–3b) = +12b
–4*(2) = –8

Combine:

5b³ – 3b² + 2b – 20b² + 12b – 8

Group:

- b³: 5b³
- b²: –3b² – 20b² = –23b²
- b: 2b + 12b = 14b
- constant: –8

Final answer for #18: 5b³ – 23b² + 14b – 8

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Final Answer:
1. x² + 9
2. 2a³ – 5a² + 11
3. 10m⁴ – 8m² + m + 6
4. 5y³ + 3y – 4
5. 7x² + 7x – 42
6. 6a² + 21a – 12
7. –2b² + 8b
8. 5k³ – 10k² – 15k
9. 6x² – 7x – 20
10. 2p² + 7p – 15
11. 4n² + 7n – 2
12. 15t² – 14t – 8
13. 6w² + 13w – 15
14. 7z² – 13z – 2
15. 2x³ + 11x² + 11x – 4
16. 3y³ – 5y² + 7y – 5
17. 8a³ + 6a² – 5a – 3
18. 5b³ – 23b² + 14b – 8