Factorization Problem: Solve with Expert Help

[Miike012]

Factroring problem...

Homework Statement

I added an attachment with questions in side... thank you.

Homework Equations

The Attempt at a Solution

 

[eumyang]

Start by writing the two polynomials in two columns:

   | 4x^3 - 3x^2 - 24x - 9 | 8x^3 - 2x^2 - 53x - 39 |

Look at the leading coefficients of the bottom-most polynomials on each column. (At the moment, there is only one polynomial in each column.) Determine how many times does one polynomial divide into the other. The left one divides into the right one twice, so put a two on the right-most side:

   | 4x^3 - 3x^2 - 24x - 9 | 8x^3 - 2x^2 - 53x - 39 | 2

Multiply that 2 by the left polynomial and put the answer underneath the right polynomial:

   | 4x^3 - 3x^2 - 24x - 9 | 8x^3 - 2x^2 - 53x - 39 | 2
   |                       | 8x^2 - 6x^2 - 48x - 18 |

Subtract:

   | 4x^3 - 3x^2 - 24x - 9 | 8x^3 - 2x^2 - 53x - 39 | 2
   |                       | 8x^2 - 6x^2 - 48x - 18 |
   |                       |------------------------|
   |                       |        4x^2 -  5x - 21 |

Repeat the process. Look at the bottom-most polynomials in each column. Determine how many times one polynomial divides into the other. The new right polynomial divides into the left polynomial x times, so write an x on the extreme left:

  x| 4x^3 - 3x^2 - 24x - 9 | 8x^3 - 2x^2 - 53x - 39 | 2
   |                       | 8x^2 - 6x^2 - 48x - 18 |
   |                       |------------------------|
   |                       |        4x^2 -  5x - 21 |

Multiply the new right polynomial by x and write underneath the left:

  x| 4x^3 - 3x^2 - 24x - 9 | 8x^3 - 2x^2 - 53x - 39 | 2
   | 4x^3 - 5x^2 - 21x     | 8x^2 - 6x^2 - 48x - 18 |
   |                       |------------------------|
   |                       |        4x^2 -  5x - 21 |

Subtract:

  x| 4x^3 - 3x^2 - 24x - 9 | 8x^3 - 2x^2 - 53x - 39 | 2
   | 4x^3 - 5x^2 - 21x     | 8x^2 - 6x^2 - 48x - 18 |
   |-----------------------|------------------------|
   |        2x^2 -  3x - 9 |        4x^2 -  5x - 21 |

Repeat the process again. Now you have two new polynomials to compare. I'm not going to go further, so hopefully you get it now.

This is method that I am not familiar with in finding the GCF between two polynomials. Anyone else have seen this?

 

[Miike012]

Thank you...
Did you already know how to do this? Or did you understand all the from just reading what I sent?

 

[Dick]

It's just the Euclidean algorithm, isn't it?

 

[Miike012]

Never hurd of it...?

 

[Dick]

Never hurd of it...?

It's the same way you find the gcd of two integers. http://en.wikipedia.org/wiki/Greatest_common_divisor_of_two_polynomials

 

[Miike012]

Interesting... My alg. books never taught me that...

 

[Dick]

Interesting... My alg. books never taught me that...

They should have. If you are doing it with polynomials, it's clearer if you practice with integers first.

 

[Miike012]

I understand the process now that I was shown how... Its just hard for me to understand why it works though...

 

[Miike012]

Will this be beneficial to know for my future in math?

 

[Dick]

Will this be beneficial to know for my future in math?

It's a fundamental thing. If your future in math depends on knowing fundamentals, then yes, this is one.