give P(x)=6x^5-13x^4-34x^3+128^2-132x=45, write p in factored for. be sure to write the full equation, including P(x)=.

Question

thought that I had it right with just factoring it out but not sure how to do it if P(x) is included 

Answer 1

If the "=45" is typed incorrectly and should have been "+45", then the polynomial factors nicely.  

 

By inspection, 1 is a root.  Divide synthetically by x-1:

 

1⌋  6     -13     -34     128     -132     45

   ______6____-7___-41_____87__-45

     6      -7      -41     87       -45       0

 

P(x) = (x-1)(6x⁴-7x³-41x²+87x-45)

 

Divide the second factor by x-1:

 

1⌋  6    -7    -41     87     -45

    ____6___-1___-42___45

     6    -1    -42     45       0

 

P(x) = (x-1)²(6x³-x²-42x+45)

 

Divide the cubic factor by x+3 synthetically:

 

-3⌋   6     -1     -42     45

    ____ -18___ 57__-45

       6    -19      15     0

 

P(x) = (x-1)²(x+3)(6x²-19x+15) = (x-1)²(x+3)(2x-3)(3x-5)

 

Answer 2

P(x) = 6x⁵ - 13x⁴ - 34x³ + 128x² - 132x - 45 = 0

 

6x⁵ - 13x⁴ - 34x³ + 128x² - 132x - 45 = 0

 

 

We can use Descartes Rule of signs to find the number possible positive, negative, and complex solutions as a guide.

 

To find the possible number of positive roots, we see how many sign changes occur in P(x).

 

The signs change 3 times.   So there are 3 possible positive roots.

 

Now we see how many sign changes occur in P(-x) to see how many possible negative roots we can have.

 

P(-x) = -6x⁵ - 13x⁴ + 34x³ + 128x² + 132x - 45

 

The signs change 2 times.  So there are 2 possible negative roots.

 

This is what our chart looks like:

 

positive  |   negative  |  complex

___________________________

   3                  2              0        ----> possibility 1

   1                  0              4        ----> possibility 2

 

This will chart along with the rational root theorem, we use synthetic division to find the zeros.   I believe this part you do on your own.

 

If there is only 1 positive root, then you should use possibility 2.  This would mean you have a linear factor multiplied by a 4th degree factor.