Printable math worksheet for practicing multiplication of binomials with multiple variables.

Sure! Let’s solve each of these multiplying binomials problems step by step. We’ll use the distributive property (FOIL method for binomials) — multiply each term in the first binomial by each term in the second binomial, then combine like terms.

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Problem 1:

(1/3 a²b + 9)(1/3 a²b + 9)

This is a square of a binomial: `(A + B)² = A² + 2AB + B²`

Let A = `1/3 a²b`, B = `9`

- A² = `(1/3 a²b)² = 1/9 a⁴b²`
- 2AB = `2 * (1/3 a²b) * 9 = 6 a²b`
- B² = `81`

✔ Answer: `1/9 a⁴b² + 6a²b + 81`

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Problem 2:

(11vw - 2)(2 + 11vw)

Notice this is the same as `(11vw - 2)(11vw + 2)` → difference of squares: `(A - B)(A + B) = A² - B²`

Let A = `11vw`, B = `2`

- A² = `(11vw)² = 121 v²w²`
- B² = `4`

✔ Answer: `121v²w² - 4`

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Problem 3:

(20rs - 8tu)(-1 - 4s)

Use distributive property:

Multiply each term in first binomial by each term in second:

- `20rs * (-1) = -20rs`
- `20rs * (-4s) = -80rs²`
- `-8tu * (-1) = 8tu`
- `-8tu * (-4s) = 32stu`

✔ Answer: `-80rs² - 20rs + 32stu + 8tu`

*(Note: Terms are written in standard order — descending powers of variables, but since multi-variable, we can group by variable combinations. No like terms to combine.)*

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Problem 4:

(-12x³ - 6x²yz)(-6yz - 3x)

Distribute each term:

First term: `-12x³ * (-6yz) = 72x³yz`
Second: `-12x³ * (-3x) = 36x⁴`
Third: `-6x²yz * (-6yz) = 36x²y²z²`
Fourth: `-6x²yz * (-3x) = 18x³yz`

Now combine like terms:

- `72x³yz + 18x³yz = 90x³yz`
- `36x⁴`
- `36x²y²z²`

✔ Answer: `36x⁴ + 90x³yz + 36x²y²z²`

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Problem 5:

(-21c⁴d⁵ - 7d³)(3c⁴d⁴ - d²)

Distribute:

- `-21c⁴d⁵ * 3c⁴d⁴ = -63c⁸d⁹`
- `-21c⁴d⁵ * (-d²) = 21c⁴d⁷`
- `-7d³ * 3c⁴d⁴ = -21c⁴d⁷`
- `-7d³ * (-d²) = 7d⁵`

Combine like terms:

- `21c⁴d⁷ - 21c⁴d⁷ = 0` → they cancel!

✔ Answer: `-63c⁸d⁹ + 7d⁵`

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Problem 6:

(m + 10n)(-4n + 2m)

Rewrite second binomial as `(2m - 4n)` for easier multiplication.

Now distribute:

- `m * 2m = 2m²`
- `m * (-4n) = -4mn`
- `10n * 2m = 20mn`
- `10n * (-4n) = -40n²`

Combine like terms:

- `-4mn + 20mn = 16mn`

✔ Answer: `2m² + 16mn - 40n²`

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Problem 7:

(-2gh + 6)(-gh + 1/2)

Distribute:

- `-2gh * (-gh) = 2g²h²`
- `-2gh * (1/2) = -gh`
- `6 * (-gh) = -6gh`
- `6 * (1/2) = 3`

Combine like terms:

- `-gh - 6gh = -7gh`

✔ Answer: `2g²h² - 7gh + 3`

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Problem 8:

(-9 + 3p⁶)(p - q⁴)

Distribute:

- `-9 * p = -9p`
- `-9 * (-q⁴) = 9q⁴`
- `3p⁶ * p = 3p⁷`
- `3p⁶ * (-q⁴) = -3p⁶q⁴`

No like terms to combine.

✔ Answer: `3p⁷ - 3p⁶q⁴ - 9p + 9q⁴`

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## ✔ Final Answers Summary:

1. `1/9 a⁴b² + 6a²b + 81`
2. `121v²w² - 4`
3. `-80rs² - 20rs + 32stu + 8tu`
4. `36x⁴ + 90x³yz + 36x²y²z²`
5. `-63c⁸d⁹ + 7d⁵`
6. `2m² + 16mn - 40n²`
7. `2g²h² - 7gh + 3`
8. `3p⁷ - 3p⁶q⁴ - 9p + 9q⁴`

Let me know if you’d like to see any step broken down further!