Polynomials practice worksheet with ten questions on polynomial properties and operations.

Let’s work through each problem one by one. These are all about multiplying polynomials — basically, using the distributive property (also called “FOIL” for binomials) to multiply terms together.

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Problem 1: 5(5p - 2)
Multiply 5 by each term inside the parentheses:
5 × 5p = 25p
5 × (-2) = -10
→ 25p - 10

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Problem 2: 6xy(x + 6y)
Distribute 6xy to both x and 6y:
6xy × x = 6x²y
6xy × 6y = 36xy²
→ 6x²y + 36xy²

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Problem 3: (q + 8)(3n + 1)
Use FOIL: First, Outer, Inner, Last
First: q × 3n = 3qn
Outer: q × 1 = q
Inner: 8 × 3n = 24n
Last: 8 × 1 = 8
Add them up: → 3qn + q + 24n + 8

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Problem 4: (4x + 2)(6x - 2)
FOIL:
First: 4x × 6x = 24x²
Outer: 4x × (-2) = -8x
Inner: 2 × 6x = 12x
Last: 2 × (-2) = -4
Combine like terms: -8x + 12x = 4x
→ 24x² + 4x - 4

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Problem 5: 6(5a + 8)
Distribute 6:
6 × 5a = 30a
6 × 8 = 48
→ 30a + 48

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Problem 6: (2a - 8b)(6a - 8b)
FOIL:
First: 2a × 6a = 12a²
Outer: 2a × (-8b) = -16ab
Inner: -8b × 6a = -48ab
Last: -8b × (-8b) = 64b²
Combine middle terms: -16ab - 48ab = -64ab
→ 12a² - 64ab + 64b²

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Problem 7: 4xy(x + 4y)
Distribute 4xy:
4xy × x = 4x²y
4xy × 4y = 16xy²
→ 4x²y + 16xy²

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Problem 8: (4a - 8b)(4a - 8b)
This is a square of a binomial: (A - B)² = A² - 2AB + B²
Or just FOIL:
First: 4a × 4a = 16a²
Outer: 4a × (-8b) = -32ab
Inner: -8b × 4a = -32ab
Last: -8b × (-8b) = 64b²
Middle: -32ab -32ab = -64ab
→ 16a² - 64ab + 64b²

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Problem 9: (x + 6)(x + 2)
FOIL:
First: x × x = x²
Outer: x × 2 = 2x
Inner: 6 × x = 6x
Last: 6 × 2 = 12
Middle: 2x + 6x = 8x
→ x² + 8x + 12

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Problem 10: (x - 8)(x - 6)
FOIL:
First: x × x = x²
Outer: x × (-6) = -6x
Inner: -8 × x = -8x
Last: -8 × (-6) = 48
Middle: -6x -8x = -14x
→ x² - 14x + 48

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Problem 11: 5(2a - 4b)
Distribute 5:
5 × 2a = 10a
5 × (-4b) = -20b
→ 10a - 20b

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Problem 12: (6p - 5)(6p + 1)
FOIL:
First: 6p × 6p = 36p²
Outer: 6p × 1 = 6p
Inner: -5 × 6p = -30p
Last: -5 × 1 = -5
Middle: 6p - 30p = -24p
→ 36p² - 24p - 5

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Final Answer:
1. 25p - 10
2. 6x²y + 36xy²
3. 3qn + q + 24n + 8
4. 24x² + 4x - 4
5. 30a + 48
6. 12a² - 64ab + 64b²
7. 4x²y + 16xy²
8. 16a² - 64ab + 64b²
9. x² + 8x + 12
10. x² - 14x + 48
11. 10a - 20b
12. 36p² - 24p - 5