Factoring Special Products worksheet - Free Printable

Let's solve each of these factoring special products problems step by step. These involve recognizing patterns like:

- Difference of squares: $ a^2 - b^2 = (a - b)(a + b) $
- Perfect square trinomials:
$ a^2 + 2ab + b^2 = (a + b)^2 $
$ a^2 - 2ab + b^2 = (a - b)^2 $
- Sum of cubes: $ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $

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1) $ x^2 - 16 $

This is a difference of squares:
$$
x^2 - 16 = x^2 - 4^2 = (x - 4)(x + 4)
$$

✔ Answer: $ (x - 4)(x + 4) $

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2) $ x^2 - 25 $

Also a difference of squares:
$$
x^2 - 25 = x^2 - 5^2 = (x - 5)(x + 5)
$$

✔ Answer: $ (x - 5)(x + 5) $

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3) $ y^2 - 2y + 1 $

Check if it’s a perfect square trinomial:
- First term: $ y^2 $
- Last term: $ 1 = 1^2 $
- Middle term: $ -2y = -2(y)(1) $

Yes! It matches $ (y - 1)^2 $

✔ Answer: $ (y - 1)^2 $

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4) $ x^2 - 8x + 16 $

Check:
- $ x^2 $
- $ 16 = 4^2 $
- Middle term: $ -8x = -2(x)(4) $

So, this is $ (x - 4)^2 $

✔ Answer: $ (x - 4)^2 $

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5) $ x^2 - 4x + 4 $

- $ x^2 $
- $ 4 = 2^2 $
- $ -4x = -2(x)(2) $

Perfect square: $ (x - 2)^2 $

✔ Answer: $ (x - 2)^2 $

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6) $ y^2 + 6y + 9 $

- $ y^2 $
- $ 9 = 3^2 $
- $ 6y = 2(y)(3) $

So, $ (y + 3)^2 $

✔ Answer: $ (y + 3)^2 $

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7) $ y^2 - 6y + 9 $

- $ y^2 $
- $ 9 = 3^2 $
- $ -6y = -2(y)(3) $

So, $ (y - 3)^2 $

✔ Answer: $ (y - 3)^2 $

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8) $ x^2 + 22x + 121 $

Check:
- $ x^2 $
- $ 121 = 11^2 $
- $ 22x = 2(x)(11) $

Perfect square: $ (x + 11)^2 $

✔ Answer: $ (x + 11)^2 $

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9) $ x^2 - 18x + 81 $

- $ x^2 $
- $ 81 = 9^2 $
- $ -18x = -2(x)(9) $

So, $ (x - 9)^2 $

✔ Answer: $ (x - 9)^2 $

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Challenge Problems

#### 10) $ x^2 + 8xy + 16y^2 $

This looks like a perfect square trinomial in terms of $ x $ and $ y $.

- $ x^2 $
- $ 16y^2 = (4y)^2 $
- Middle term: $ 8xy = 2(x)(4y) $

So, $ (x + 4y)^2 $

✔ Answer: $ (x + 4y)^2 $

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#### 11) $ x^3 + 8 $

This is a sum of cubes:
$$
x^3 + 8 = x^3 + 2^3 = (x + 2)(x^2 - 2x + 4)
$$

Use formula: $ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $

✔ Answer: $ (x + 2)(x^2 - 2x + 4) $

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#### 12) $ y^2 - 42y + 441 $

Check:
- $ y^2 $
- $ 441 = 21^2 $
- $ -42y = -2(y)(21) $

Perfect square: $ (y - 21)^2 $

✔ Answer: $ (y - 21)^2 $

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#### 13) $ x^2 - 10x - 25 $

Wait — this is not a perfect square trinomial because:
- $ x^2 $
- $ -25 $ → negative, so can't be a square of real number
- But $ -10x $ is not matching the pattern of perfect square

Let’s check: Is it factorable?

We need two numbers that multiply to $ -25 $ and add to $ -10 $

Try:
- $ 5 $ and $ -5 $: $ 5 \times -5 = -25 $, but $ 5 + (-5) = 0 $ → no
- $ 25 $ and $ -1 $: sum = 24 → no
- $ -25 $ and $ 1 $: sum = -24 → no

No integer pairs work. So, this does NOT factor nicely over integers.

But wait — let’s double-check the problem:
Is it $ x^2 - 10x - 25 $? Yes.

It is not a special product.

But maybe there's a typo? Because $ x^2 - 10x + 25 = (x - 5)^2 $

But as written: $ x^2 - 10x - 25 $ → does not factor using integers.

However, if we use the quadratic formula:

$$
x = \frac{10 \pm \sqrt{(-10)^2 - 4(1)(-25)}}{2} = \frac{10 \pm \sqrt{100 + 100}}{2} = \frac{10 \pm \sqrt{200}}{2}
= \frac{10 \pm 10\sqrt{2}}{2} = 5 \pm 5\sqrt{2}
$$

So, factors are:
$$
(x - (5 + 5\sqrt{2}))(x - (5 - 5\sqrt{2}))
$$

But since the problem asks for special products, and this isn’t one, likely it’s not factorable with rational coefficients.

✔ Answer: Does not factor over integers (or cannot be factored as a special product)

But perhaps there was a typo — maybe it was meant to be $ x^2 - 10x + 25 $? That would be $ (x - 5)^2 $

But based on what’s written: $ x^2 - 10x - 25 $ → cannot be factored using special product rules.

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✔ Final Answers:

1. $ (x - 4)(x + 4) $
2. $ (x - 5)(x + 5) $
3. $ (y - 1)^2 $
4. $ (x - 4)^2 $
5. $ (x - 2)^2 $
6. $ (y + 3)^2 $
7. $ (y - 3)^2 $
8. $ (x + 11)^2 $
9. $ (x - 9)^2 $
10. $ (x + 4y)^2 $
11. $ (x + 2)(x^2 - 2x + 4) $
12. $ (y - 21)^2 $
13. Not factorable as a special product (no rational factors)

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Let me know if you'd like the worksheet completed neatly or want explanations for any specific one!