The document provides a math prayer asking God for guidance in learning about factoring polynomials. It then gives a table with examples of factoring various polynomials by finding the greatest common monomial factor (GCMF). The GCMF is used to factor each polynomial into the product of the GCMF and a polynomial quotient. Examples include factoring 6m+8, 27d4o5t3a6-18d2o3t6-15d6o4, and 12x3y5-20x5y2z. The document concludes with additional examples of factorized expressions and their relationships.

1. Common Monomial Factoring

2. Math Prayer
Lord,
You have created us in different shapes and angles
But still you made sure that we are congruent one way or another.
No matter which sides, even if there are terms undefined, your
teachings will remain our guide. May our lesson today maybe made
parallel with what you want us to become. And by the time our paths
perpendicularly crossed, may we face you with confidence
That you will shower us with your unending love and providence.
AMEN.

3. Complete the table below:
Polynomial
Greatest
Common
Monomial Factor
(GCMF)
Quotient of Polynomial
and GCMF
Factored Form
6m + 8 2 3m + 4 2(3m +4)
27d4o5t3a6 – 18d2o3t6 – 15d6o4 9d2o2t3a6 – 6t6 – 5d4o
4mo2 4mo2(3m + o)
4(12) + 4(8) 4
12WI3N5 – 16WIN + 20WINNER
12x3y5 – 20x5y2z 4x3y2
12x5y4 – 16x3y4 + 28x6 (3x2y4 – 4y4 + 7x3)
8x2 + 16x 8x
3xy2 3xy2 (2x + y – 1)
3 3(s3 + 2e4 – 3o2 – 4h)

4. Complete the table below:
Polynomial
Greatest Common
Monomial Factor
(GCMF)
Quotient of Polynomial and
GCMF
Factored Form
6m + 8 2 3m + 4 2(3m +4)
27d4o5t3a6 – 18d2o3t6 – 15d6o4 3d2o3 9d2o2t3a6 – 6t6 – 5d4o
3d2o3(9d2o2t3a6 – 6t6 –
5d4o)
4mo2 4mo2(3m + o)
4(12) + 4(8) 4
12WI3N5 – 16WIN + 20WINNER
12x3y5 – 20x5y2z 4x3y2
12x5y4 – 16x3y4 + 28x6 (3x2y4 – 4y4 + 7x3)
8x2 + 16x 8x
3xy2 3xy2 (2x + y – 1)
3 3(s3 + 2e4 – 3o2 – 4h)

5. Complete the table below:
Polynomial
Greatest Common
Monomial Factor
(GCMF)
Quotient of Polynomial and
GCMF
Factored Form
6m + 8 2 3m + 4 2(3m +4)
27d4o5t3a6 – 18d2o3t6 – 15d6o4 3d2o3 9d2o2t3a6 – 6t6 – 5d4o
3d2o3(9d2o2t3a6 – 6t6 –
5d4o)
12m2o2 + 4mo3 4mo2 3m + o 4mo2(3m + o)
4(12) + 4(8) 4
12WI3N5 – 16WIN + 20WINNER
12x3y5 – 20x5y2z 4x3y2
12x5y4 – 16x3y4 + 28x6 (3x2y4 – 4y4 + 7x3)
8x2 + 16x 8x
3xy2 3xy2 (2x + y – 1)
3 3(s3 + 2e4 – 3o2 – 4h)

6. Complete the table below:
Polynomial
Greatest Common
Monomial Factor
(GCMF)
Quotient of Polynomial and
GCMF
Factored Form
6m + 8 2 3m + 4 2(3m +4)
27d4o5t3a6 – 18d2o3t6 – 15d6o4 3d2o3 9d2o2t3a6 – 6t6 – 5d4o
3d2o3(9d2o2t3a6 – 6t6 –
5d4o)
12m2o2 + 4mo3 4mo2 3m + o 4mo2(3m + o)
4(12) + 4(8) 4 12 + 8 4(12 + 8)
12WI3N5 – 16WIN + 20WINNER
12x3y5 – 20x5y2z 4x3y2
12x5y4 – 16x3y4 + 28x6 (3x2y4 – 4y4 + 7x3)
8x2 + 16x 8x
3xy2 3xy2 (2x + y – 1)
3 3(s3 + 2e4 – 3o2 – 4h)

7. Complete the table below:
Polynomial
Greatest Common
Monomial Factor
(GCMF)
Quotient of Polynomial and
GCMF
Factored Form
6m + 8 2 3m + 4 2(3m +4)
27d4o5t3a6 – 18d2o3t6 – 15d6o4 3d2o3 9d2o2t3a6 – 6t6 – 5d4o
3d2o3(9d2o2t3a6 – 6t6 –
5d4o)
12m2o2 + 4mo3 4mo2 3m + o 4mo2(3m + o)
4(12) + 4(8) 4 12 + 8 4(12 + 8)
12WI3N5 – 16WIN + 20WINNER 4WIN 3I2N4 – 4 + 5NER 4WIN(3I2N4 – 4 + 5NER)
12x3y5 – 20x5y2z 4x3y2
12x5y4 – 16x3y4 + 28x6 (3x2y4 – 4y4 + 7x3)
8x2 + 16x 8x
3xy2 3xy2 (2x + y – 1)
3 3(s3 + 2e4 – 3o2 – 4h)

8. Complete the table below:
Polynomial
Greatest Common
Monomial Factor
(GCMF)
Quotient of Polynomial and
GCMF
Factored Form
6m + 8 2 3m + 4 2(3m +4)
27d4o5t3a6 – 18d2o3t6 – 15d6o4 3d2o3 9d2o2t3a6 – 6t6 – 5d4o
3d2o3(9d2o2t3a6 – 6t6 –
5d4o)
12m2o2 + 4mo3 4mo2 3m + o 4mo2(3m + o)
4(12) + 4(8) 4 12 + 8 4(12 + 8)
12WI3N5 – 16WIN + 20WINNER 4WIN 3I2N4 – 4 + 5NER 4WIN(3I2N4 – 4 + 5NER)
12x3y5 – 20x5y2z 4x3y2 3y2 – 5x2z 4x3y2(3y2 – 5x2z)
12x5y4 – 16x3y4 + 28x6 (3x2y4 – 4y4 + 7x3)
8x2 + 16x 8x
3xy2 3xy2 (2x + y – 1)
3 3(s3 + 2e4 – 3o2 – 4h)

9. Complete the table below:
Polynomial
Greatest Common
Monomial Factor
(GCMF)
Quotient of Polynomial and
GCMF
Factored Form
6m + 8 2 3m + 4 2(3m +4)
27d4o5t3a6 – 18d2o3t6 – 15d6o4 3d2o3 9d2o2t3a6 – 6t6 – 5d4o
3d2o3(9d2o2t3a6 – 6t6 –
5d4o)
12m2o2 + 4mo3 4mo2 3m + o 4mo2(3m + o)
4(12) + 4(8) 4 12 + 8 4(12 + 8)
12WI3N5 – 16WIN + 20WINNER 4WIN 3I2N4 – 4 + 5NER 4WIN(3I2N4 – 4 + 5NER)
12x3y5 – 20x5y2z 4x3y2 3y2 – 5x2z 4x3y2(3y2 – 5x2z)
12x5y4 – 16x3y4 + 28x6 4x3 (3x2y4 – 4y4 + 7x3) 4x3(3x2y4 – 4y4 + 7x3)
8x2 + 16x 8x
3xy2 3xy2 (2x + y – 1)
3 3(s3 + 2e4 – 3o2 – 4h)

10. Complete the table below:
Polynomial
Greatest Common
Monomial Factor
(GCMF)
Quotient of Polynomial and
GCMF
Factored Form
6m + 8 2 3m + 4 2(3m +4)
27d4o5t3a6 – 18d2o3t6 – 15d6o4 3d2o3 9d2o2t3a6 – 6t6 – 5d4o
3d2o3(9d2o2t3a6 – 6t6 –
5d4o)
12m2o2 + 4mo3 4mo2 3m + o 4mo2(3m + o)
4(12) + 4(8) 4 12 + 8 4(12 + 8)
12WI3N5 – 16WIN + 20WINNER 4WIN 3I2N4 – 4 + 5NER 4WIN(3I2N4 – 4 + 5NER)
12x3y5 – 20x5y2z 4x3y2 3y2 – 5x2z 4x3y2(3y2 – 5x2z)
12x5y4 – 16x3y4 + 28x6 4x3 (3x2y4 – 4y4 + 7x3) 4x3(3x2y4 – 4y4 + 7x3)
8x2 + 16x 8x x + 2 8x(x + 2)
3xy2 3xy2 (2x + y – 1)
3 3(s3 + 2e4 – 3o2 – 4h)

11. Complete the table below:
Polynomial
Greatest Common
Monomial Factor
(GCMF)
Quotient of Polynomial and
GCMF
Factored Form
6m + 8 2 3m + 4 2(3m +4)
27d4o5t3a6 – 18d2o3t6 – 15d6o4 3d2o3 9d2o2t3a6 – 6t6 – 5d4o
3d2o3(9d2o2t3a6 – 6t6 –
5d4o)
12m2o2 + 4mo3 4mo2 3m + o 4mo2(3m + o)
4(12) + 4(8) 4 12 + 8 4(12 + 8)
12WI3N5 – 16WIN + 20WINNER 4WIN 3I2N4 – 4 + 5NER 4WIN(3I2N4 – 4 + 5NER)
12x3y5 – 20x5y2z 4x3y2 3y2 – 5x2z 4x3y2(3y2 – 5x2z)
12x5y4 – 16x3y4 + 28x6 4x3 (3x2y4 – 4y4 + 7x3) 4x3(3x2y4 – 4y4 + 7x3)
8x2 + 16x 8x x + 2 8x(x + 2)
6x2y2 + 3xy3 -3xy2 3xy2 2x + y - 1 3xy2 (2x + y – 1)
3 3(s3 + 2e4 – 3o2 – 4h)

12. Complete the table below:
Polynomial
Greatest Common
Monomial Factor
(GCMF)
Quotient of Polynomial and
GCMF
Factored Form
6m + 8 2 3m + 4 2(3m +4)
27d4o5t3a6 – 18d2o3t6 – 15d6o4 3d2o3 9d2o2t3a6 – 6t6 – 5d4 3d2o3(9d2o2t3a6 – 6t6 –
5d4o)
12m2o2 + 4mo3 4mo2 3m + o 4mo2(3m + o)
4(12) + 4(8) 4 12 + 8 4(12 + 8)
12WI3N5 – 16WIN + 20WINNER 4WIN 3I2N4 – 4 + 5NER 4WIN(3I2N4 – 4 + 5NER)
12x3y5 – 20x5y2z 4x3y2 3y2 – 5x2z 4x3y2(3y2 – 5x2z)
12x5y4 – 16x3y4 + 28x6 4x3 (3x2y4 – 4y4 + 7x3) 4x3(3x2y4 – 4y4 + 7x3)
8x2 + 16x 8x x + 2 8x(x + 2)
6x2y2 + 3xy3 -3xy2 3xy2 2x + y - 1 3xy2 (2x + y – 1)
3s3 + 6e4 – 9o2 – 12h 3 s3 + 2e4 – 3o2 – 4h 3(s3 + 2e4 – 3o2 – 4h)

13. Observe how the expressions below are factored and observe
the relationships of the term with each other
 x2 – y2 = (x + y)(x – y)
 4x2 – 36 = (2x + 6)(2x – 6)
 a2b4 – 81 = (ab2 + 9)(ab2 – 9)