Factoring the greatest common monomial factor | PPT - Free Printable

Let's solve each problem step by step. The task is to factor out a monomial from each expression. If the resulting polynomial is prime (cannot be factored further), we write "prime".

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Problem 8:

Expression: $ 12x^2 - 9x + 15 $

Step 1: Look for the greatest common factor (GCF) of the coefficients:
- Coefficients: 12, 9, 15
- GCF of 12, 9, 15 is 3

Now factor out 3:

$$
12x^2 - 9x + 15 = 3(4x^2 - 3x + 5)
$$

Now check if $ 4x^2 - 3x + 5 $ can be factored.

Try factoring:
We look for two numbers that multiply to $ 4 \times 5 = 20 $ and add to $-3$.
Possible pairs: (–1, –20), (–2, –10), (–4, –5), etc. None add to –3.
So it does not factor over integers.

✔ So, $ 4x^2 - 3x + 5 $ is prime.

Answer: $ \boxed{3(4x^2 - 3x + 5)} $

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Problem 9:

Expression: $ 3n^3 - 12n^2 - 30n $

Step 1: Find GCF of coefficients: 3, 12, 30 → GCF is 3
Also, all terms have at least one $ n $ → GCF includes $ 3n $

Factor out $ 3n $:

$$
3n^3 - 12n^2 - 30n = 3n(n^2 - 4n - 10)
$$

Now check if $ n^2 - 4n - 10 $ factors.

We need two numbers that multiply to $-10$ and add to $-4$:
Try: (–1, 10), (–2, 5), (–5, 2), (–10, 1) → none add to –4
No integer pair works → prime

Answer: $ \boxed{3n(n^2 - 4n - 10)} $

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Problem 10:

Expression: $ 9n^2 - 4n + 12 $

Check for GCF: 9, 4, 12 → GCF is 1 (no common factor)

So no monomial to factor out. But let’s see if it factors at all.

Try factoring $ 9n^2 - 4n + 12 $

We want two numbers multiplying to $ 9 \times 12 = 108 $, adding to $-4$

Try: (–6, –18): product 108, sum –24
(–9, –12): 108, sum –21
(–3, –36): 108, sum –39
None add to –4 → no factoring possible.

So this quadratic is prime.

Answer: $ \boxed{\text{prime}} $

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Problem 11:

Expression: $ 2x^3 - 3x^2 + 5x $

All terms have $ x $, so factor out $ x $:

$$
2x^3 - 3x^2 + 5x = x(2x^2 - 3x + 5)
$$

Now check $ 2x^2 - 3x + 5 $

Find two numbers multiplying to $ 2 \times 5 = 10 $, adding to $-3$:
(–1, –10): sum –11
(–2, –5): sum –7
(–5, –2): sum –7
No pair adds to –3 → prime

Answer: $ \boxed{x(2x^2 - 3x + 5)} $

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Problem 12:

Expression: $ 13m + 26m^2 - 39m^3 $

Rewrite in descending order:
$ -39m^3 + 26m^2 + 13m $

Look for GCF:
Coefficients: 13, 26, 39 → GCF is 13
All terms have $ m $ → GCF is $ 13m $

Factor out $ 13m $:

$$
13m + 26m^2 - 39m^3 = 13m(1 + 2m - 3m^2)
$$

But better to write in standard form:

$$
= 13m(-3m^2 + 2m + 1)
$$

Or rearrange inside:

$$
= 13m(-3m^2 + 2m + 1)
$$

We can factor out the negative sign for nicer form:

$$
= -13m(3m^2 - 2m - 1)
$$

Now factor $ 3m^2 - 2m - 1 $

We need two numbers multiplying to $ 3 \times (-1) = -3 $, adding to $-2$:
→ $-3$ and $+1$: $-3 + 1 = -2$, $-3 \times 1 = -3$

So split middle term:

$$
3m^2 - 3m + m - 1 = 3m(m - 1) + 1(m - 1) = (3m + 1)(m - 1)
$$

So:

$$
-13m(3m^2 - 2m - 1) = -13m(3m + 1)(m - 1)
$$

But the original question says “factor out a monomial”, so we only need to factor out the monomial, not fully factor.

So just factoring out $ 13m $:

$$
13m + 26m^2 - 39m^3 = 13m(1 + 2m - 3m^2)
$$

But usually we write in descending powers:

$$
= 13m(-3m^2 + 2m + 1)
$$

Alternatively, factor out $-13m$ to make leading coefficient positive:

$$
= -13m(3m^2 - 2m - 1)
$$

But since the instruction is to factor out a monomial, any correct monomial factor is acceptable. Most natural is:

$$
\boxed{13m(-3m^2 + 2m + 1)}
$$

Alternatively, you might write:

$$
\boxed{-13m(3m^2 - 2m - 1)}
$$

But since the original expression starts with positive $13m$, perhaps keep as:

$$
\boxed{13m(1 + 2m - 3m^2)}
$$

However, standard form is descending powers.

✔ Final answer: $ \boxed{13m(-3m^2 + 2m + 1)} $

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Problem 13:

Expression: $ 17x^2 + 34x + 51 $

GCF of 17, 34, 51:
17 divides all: $ 17 \times 1 = 17 $, $ 17 \times 2 = 34 $, $ 17 \times 3 = 51 $

So GCF is 17

Factor out 17:

$$
17x^2 + 34x + 51 = 17(x^2 + 2x + 3)
$$

Now check $ x^2 + 2x + 3 $:
Need two numbers multiplying to 3, adding to 2:
(1,3): sum 4 → too big
(–1,–3): sum –4 → no
No such integers → prime

Answer: $ \boxed{17(x^2 + 2x + 3)} $

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Problem 14:

Expression: $ 18m^2n^4 - 12m^2n^3 + 24m^2n^2 $

Look at each term:
- Coefficients: 18, 12, 24 → GCF is 6
- Variables: All have $ m^2 $, $ n^2 $
- Lowest power of $ n $: $ n^2 $

So GCF is $ 6m^2n^2 $

Factor that out:

$$
18m^2n^4 - 12m^2n^3 + 24m^2n^2 = 6m^2n^2(3n^2 - 2n + 4)
$$

Now check $ 3n^2 - 2n + 4 $

Need two numbers multiplying to $ 3 \times 4 = 12 $, adding to $-2$:
(–1, –12): sum –13
(–2, –6): sum –8
(–3, –4): sum –7
(–4, –3): sum –7
No pair adds to –2 → prime

Answer: $ \boxed{6m^2n^2(3n^2 - 2n + 4)} $

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 8 | $ 3(4x^2 - 3x + 5) $ |
| 9 | $ 3n(n^2 - 4n - 10) $ |
| 10 | prime |
| 11 | $ x(2x^2 - 3x + 5) $ |
| 12 | $ 13m(-3m^2 + 2m + 1) $ or $ -13m(3m^2 - 2m - 1) $ |
| 13 | $ 17(x^2 + 2x + 3) $ |
| 14 | $ 6m^2n^2(3n^2 - 2n + 4) $ |

> Note: For problem 12, either form is acceptable depending on preference, but factoring out $ 13m $ is sufficient per instructions.

Let me know if you'd like a visual explanation or help with checking factoring!