Algebra 1 Worksheets | Monomials and Polynomials Worksheets - Free Printable

Let's solve each of the factoring problems step by step. The goal is to factor each expression completely. If it cannot be factored over the integers, we write "Non-factorable".

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1) $ p^2 - 31 $

This is a difference of squares only if 31 is a perfect square. But $ \sqrt{31} \approx 5.57 $, not an integer.

So:
$ p^2 - 31 $ → Not factorable over integers
✔ Answer: Non-factorable

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2) $ y^4 - 10y^3 + 5y^2 $

First, factor out the greatest common factor (GCF):

All terms have $ y^2 $:
$ y^2(y^2 - 10y + 5) $

Now check if the quadratic $ y^2 - 10y + 5 $ factors.

Look for two numbers that multiply to 5 and add to -10.
Possible pairs: (1,5), (-1,-5) — none add to -10.
Discriminant: $ (-10)^2 - 4(1)(5) = 100 - 20 = 80 $ → not a perfect square → doesn't factor nicely.

So, only factor is $ y^2 $.

✔ Answer: $ y^2(y^2 - 10y + 5) $

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3) $ g^4 - 4g^2 $

Factor out GCF: $ g^2 $

$ g^2(g^2 - 4) $

Now $ g^2 - 4 $ is a difference of squares:
$ g^2 - 4 = (g - 2)(g + 2) $

So full factorization:
$ g^2(g - 2)(g + 2) $

✔ Answer: $ g^2(g - 2)(g + 2) $

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4) $ 75h^4 + 90h^3 + 27h^2 $

Factor out GCF: all terms divisible by $ 3h^2 $

$ 3h^2(25h^2 + 30h + 9) $

Now factor $ 25h^2 + 30h + 9 $

Check:
Try to factor as $ (5h + a)(5h + b) $

We need $ a \cdot b = 9 $, and $ 5b + 5a = 30 $ → $ a + b = 6 $

So $ a = 3, b = 3 $

→ $ (5h + 3)(5h + 3) = (5h + 3)^2 $

So full factorization:
$ 3h^2(5h + 3)^2 $

✔ Answer: $ 3h^2(5h + 3)^2 $

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5) $ 16b^2 - 25 $

Difference of squares:

$ 16b^2 = (4b)^2 $, $ 25 = 5^2 $

So:
$ (4b)^2 - 5^2 = (4b - 5)(4b + 5) $

✔ Answer: $ (4b - 5)(4b + 5) $

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6) $ 9k^4 - 49k^2 $

Factor out GCF: $ k^2 $

$ k^2(9k^2 - 49) $

Now $ 9k^2 - 49 $ is difference of squares:

$ (3k)^2 - 7^2 = (3k - 7)(3k + 7) $

So full factorization:
$ k^2(3k - 7)(3k + 7) $

✔ Answer: $ k^2(3k - 7)(3k + 7) $

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7) $ g^2 - 14g + 49 $

This is a perfect square trinomial:

Check:
$ g^2 - 14g + 49 = (g - 7)^2 $

Because: $ (g - 7)^2 = g^2 - 14g + 49 $

✔ Answer: $ (g - 7)^2 $

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8) $ 9p^4 - 30p^3 + 25p^2 $

Factor out GCF: $ p^2 $

$ p^2(9p^2 - 30p + 25) $

Now factor $ 9p^2 - 30p + 25 $

Check if perfect square trinomial:

- $ 9p^2 = (3p)^2 $
- $ 25 = 5^2 $
- Middle term: $ 2 \cdot 3p \cdot 5 = 30p $ → yes!

So:
$ (3p - 5)^2 $

Thus:
$ p^2(3p - 5)^2 $

✔ Answer: $ p^2(3p - 5)^2 $

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9) $ c^4 + 18c^3 + 2c^2 $

Factor out GCF: $ c^2 $

$ c^2(c^2 + 18c + 2) $

Now check if $ c^2 + 18c + 2 $ factors.

Need two numbers that multiply to 2 and add to 18:
Possible pairs: (1,2), (-1,-2) — none add to 18.

Discriminant: $ 18^2 - 4(1)(2) = 324 - 8 = 316 $ → not a perfect square → no rational roots.

So, cannot factor further.

✔ Answer: $ c^2(c^2 + 18c + 2) $

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10) $ y^4 - 49y^2 $

Factor out GCF: $ y^2 $

$ y^2(y^2 - 49) $

Now $ y^2 - 49 $ is difference of squares:
$ y^2 - 7^2 = (y - 7)(y + 7) $

So:
$ y^2(y - 7)(y + 7) $

✔ Answer: $ y^2(y - 7)(y + 7) $

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✔ Final Answers:

1) Non-factorable
2) $ y^2(y^2 - 10y + 5) $
3) $ g^2(g - 2)(g + 2) $
4) $ 3h^2(5h + 3)^2 $
5) $ (4b - 5)(4b + 5) $
6) $ k^2(3k - 7)(3k + 7) $
7) $ (g - 7)^2 $
8) $ p^2(3p - 5)^2 $
9) $ c^2(c^2 + 18c + 2) $
10) $ y^2(y - 7)(y + 7) $

Let me know if you'd like these written neatly in a formatted sheet!