Factoring the Common Monomial Factor Worksheet | PDF - Free Printable

1. 8a + 10a² = 2a(4 + 5a)
2. 6a² - 9a = 3a(2a - 3)
3. 16p³ + 4p² = 4p²(4p + 1)
4. 25m² + 5m = 5m(5m + 1)
5. 15n² + 12n = 3n(5n + 4)
6. 20x² + 5x² = 5x²(4 + 1) → 5x²(5) → 25x² (Note: This expression simplifies directly to 25x²; factoring out 5x² gives 5x²(4 + 1), which is 5x²·5, so the factored form is 5x²·5 or simply 25x². However, if we treat it as written, 20x² + 5x² = 25x², and since it's a monomial, it doesn't have a common factor other than itself. But if the intent was to factor 20x² + 5x, then it would be 5x(4x + 1). Given the problem, I'll assume it's correctly written as 20x² + 5x², so the factored form is 5x²(4 + 1) = 5x²·5, but typically we write it as 25x². For consistency with factoring, perhaps it's a typo and should be 20x² + 5x. I'll provide both: If 20x² + 5x², then 25x². If 20x² + 5x, then 5x(4x + 1). Since the problem says "factor", and 25x² is already simplified, I'll note that. But looking at the pattern, likely it's 20x² + 5x. So I'll go with 5x(4x + 1). Actually, rechecking: 20x² + 5x² = 25x², which is a single term, so no factoring needed beyond recognizing it's 25x². But the instruction is to factor, so perhaps it's intended as 20x² + 5x. To avoid confusion, I'll factor as is: 20x² + 5x² = 5x²(4 + 1) = 5x²·5. But that's not standard. Better to say: 20x² + 5x² = 25x², and since it's a monomial, the greatest common factor is 25x² itself. But that seems odd. Perhaps it's a mistake. I'll assume it's 20x² + 5x for factoring purpose. So: 5x(4x + 1). I think that's what was intended.
7. 15y³ - 3y² = 3y²(5y - 1)
8. a + 10b = 1(a + 10b) [No common factor other than 1]
9. 6b² + 10b² = 16b² = 2b²(8) or 16b², but factoring: 2b²(8) is not standard; better to say 16b². Or if we factor out 2, 2(8b²). But typically, we factor out the GCF, which is 2b²? No, 6b² + 10b² = 16b², so GCF is 16b². But again, it's a monomial. Perhaps intended as 6b² + 10b. Then 2b(3b + 5). I'll assume that. So: 2b(3b + 5)
10. 25p + 10 = 5(5p + 2)
11. 4y² + 8y = 4y(y + 2)
12. 32x² + 8ab - 80b² = 8(4x² + ab - 10b²)
13. 6x²y + 2xy² - 6y = 2y(3x² + xy - 3)
14. 20x²y² + 4x²y - 20xy² = 4xy(5xy + x - 5y)
15. 32mn² - 40mn + 24 = 8(4mn² - 5mn + 3)
16. 8c²d² - 4cd - 3 = ? Let's see: factors of 8*3=24, and -4. Possible: (4cd - 3)(2cd + 1)? 4cd*2cd=8c²d², 4cd*1=4cd, -3*2cd=-6cd, -3*1=-3. So 8c²d² -2cd -3, not matching. (8cd - 3)(cd + 1) = 8c²d² +8cd -3cd -3 = 8c²d² +5cd -3, not good. Perhaps no rational factors. But let's check GCF: coefficients 8,4,3, GCF is 1. So perhaps it's prime. But maybe I missed something. 8c²d² - 4cd - 3. Let me try to factor: set u=cd, then 8u² -4u -3. Discriminant 16 + 96 = 112, not square, so no rational factors. So it might be left as is, or perhaps there's a typo. I'll leave it as 8c²d² - 4cd - 3, but since it's to factor, and no common factor, perhaps it's irreducible. But let's move on.
17. 12x² - 15x²y² + 15y² = 3(4x² - 5x²y² + 5y²) — wait, 12x² -15x²y² +15y². Factor out 3: 3(4x² -5x²y² +5y²). Can we factor more? 4x² -5x²y² +5y² = x²(4 -5y²) +5y², not obvious. Perhaps group: but no common factor. So 3(4x² -5x²y² +5y²). Or factor as is.
18. 21m²n² + 27mn² + 12m²n = 3mn(7mn + 9n + 4m) — let's check: 3mn *7mn=21m²n², 3mn*9n=27mn², 3mn*4m=12m²n, yes.
19. 4x²y - 20xy²z - 24x²y² + 20x³yz = 4xy(x - 5yz - 6xy + 5x²z) — let's factor: GCF of coefficients 4,20,24,20 is 4. Variables: x,y,z. Each term has at least x and y. So 4xy(x - 5yz - 6xy + 5x²z). We can write as 4xy( x + 5x²z - 6xy - 5yz ). But perhaps rearrange: 4xy(5x²z - 6xy + x - 5yz). Not easily factorable further.
20. 90x²y²z - 18x²y³z - 27x²y² = 9x²y²(10z - 2yz - 3) — let's see: 9x²y² *10z=90x²y²z, 9x²y²*(-2yz)= -18x²y³z, 9x²y²*(-3)= -27x²y², yes.
21. 55mpq + 5m²pq + 15mp² + 25mnp = 5mp(11q + mq + 3p + 5n) — check: 5mp*11q=55mpq, 5mp*mq=5m²pq, 5mp*3p=15mp², 5mp*5n=25mnp, yes.
22. 18p + 60mp + 72p² - 24mp² = 6p(3 + 10m + 12p - 4mp) — factor out 6p: 6p*3=18p, 6p*10m=60mp, 6p*12p=72p², 6p*(-4mp)= -24mp², yes. We can write as 6p(12p - 4mp + 10m + 3).