Factoring_Polynomials - Free Printable

The image shows a list of 14 algebraic expressions that are already in their factored form. Since no specific instruction is given (like “expand” or “simplify”), and based on typical homework contexts, the most likely task is to expand each expression — that is, multiply out the factors to write them as polynomials.

Let’s go through each one step by step, expanding carefully.

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1. 5(x + 1)(x – 1)
First, notice (x + 1)(x – 1) is a difference of squares: x² – 1
Then multiply by 5:
→ 5(x² – 1) = 5x² – 5

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2. (x + 5)(x – 2)
Use FOIL:
x·x = x²
x·(-2) = -2x
5·x = 5x
5·(-2) = -10
Combine: x² + (-2x + 5x) – 10 = x² + 3x – 10

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3. (x + 2)(2x – 1)
FOIL:
x·2x = 2x²
x·(-1) = -x
2·2x = 4x
2·(-1) = -2
Combine: 2x² + (-x + 4x) – 2 = 2x² + 3x – 2

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4. x(x + 2)²
First expand (x + 2)² = x² + 4x + 4
Then multiply by x:
x·(x² + 4x + 4) = x³ + 4x² + 4x

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5. 2(5 – x)(5 + x)
Notice (5 – x)(5 + x) = 25 – x² (difference of squares)
Multiply by 2:
→ 2(25 – x²) = 50 – 2x²

*(Note: Usually we write highest degree first, so –2x² + 50 is also acceptable, but 50 – 2x² is fine too.)*

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6. 2(2a – b)(x – 3)
This has two different variables: a and x. We can’t combine terms across variables.
First multiply (2a – b)(x – 3):
= 2a·x + 2a·(-3) + (-b)·x + (-b)·(-3)
= 2ax – 6a – bx + 3b
Now multiply by 2:
→ 2(2ax – 6a – bx + 3b) = 4ax – 12a – 2bx + 6b

We can rearrange if needed, but this is expanded.

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7. (x – 2)(x² + 2x + 4)
This looks like the factorization of a difference of cubes:
Recall: (x – 2)(x² + 2x + 4) = x³ – 8
Let’s verify by multiplying:

x·x² = x³
x·2x = 2x²
x·4 = 4x
-2·x² = -2x²
-2·2x = -4x
-2·4 = -8

Combine:
x³ + (2x² – 2x²) + (4x – 4x) – 8 = x³ – 8

✔ Correct.

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8. 6(y – 2)(y – 6)
First multiply (y – 2)(y – 6):
= y² – 6y – 2y + 12 = y² – 8y + 12
Now multiply by 6:
→ 6y² – 48y + 72

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9. 5(2a – 3b)(a + b)
First multiply (2a – 3b)(a + b):
= 2a·a + 2a·b – 3b·a – 3b·b
= 2a² + 2ab – 3ab – 3b²
= 2a² – ab – 3b²
Now multiply by 5:
→ 10a² – 5ab – 15b²

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10. (a² + 4)(a + 2)(a – 2)
Notice (a + 2)(a – 2) = a² – 4
So now: (a² + 4)(a² – 4) = (a²)² – (4)² = a⁴ – 16
*(Difference of squares again!)*
→ a⁴ – 16

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11. (x – 2)(x + 1)(x – 1)
First multiply (x + 1)(x – 1) = x² – 1
Then multiply by (x – 2):
(x – 2)(x² – 1) = x·x² + x·(-1) –2·x² –2·(-1)
= x³ – x – 2x² + 2
Rearrange: x³ – 2x² – x + 2

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12. -3x²y(5x – 4)(3x + 2)
First multiply (5x – 4)(3x + 2):
= 5x·3x + 5x·2 –4·3x –4·2
= 15x² + 10x – 12x – 8
= 15x² – 2x – 8
Now multiply by -3x²y:
= -3x²y · (15x² – 2x – 8)
= -3x²y·15x² + (-3x²y)·(-2x) + (-3x²y)·(-8)
= -45x⁴y + 6x³y + 24x²y

→ –45x⁴y + 6x³y + 24x²y

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13. b²(a + 3)²
First expand (a + 3)² = a² + 6a + 9
Then multiply by b²:
→ b²(a² + 6a + 9) = a²b² + 6ab² + 9b²

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14. 2a(3a + 2)(9a² – 6a + 4)
Notice: (3a + 2)(9a² – 6a + 4) is the expansion of a sum of cubes:
Recall: (A + B)(A² – AB + B²) = A³ + B³
Here, A = 3a, B = 2 → (3a)³ + (2)³ = 27a³ + 8
Check by multiplying:

(3a + 2)(9a² – 6a + 4)
= 3a·9a² = 27a³
3a·(-6a) = -18a²
3a·4 = 12a
2·9a² = 18a²
2·(-6a) = -12a
2·4 = 8

Combine:
27a³ + (-18a² + 18a²) + (12a – 12a) + 8 = 27a³ + 8 ✔

Now multiply by 2a:
→ 2a(27a³ + 8) = 54a + 16a

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## Final Answer:

1. 5x² – 5
2. x² + 3x – 10
3. 2x² + 3x – 2
4. x³ + 4x² + 4x
5. 50 – 2x²
6. 4ax – 12a – 2bx + 6b
7. x³ – 8
8. 6y² – 48y + 72
9. 10a² – 5ab – 15b²
10. a⁴ – 16
11. x³ – 2x² – x + 2
12. –45xy + 6x³y + 24x²y
13. a²b² + 6ab² + 9b²
14. 54a⁴ + 16a