logo for solving-math-problems.com
leftimage for solving-math-problems.com

Solve the polynomial x⁷-4x⁶-x⁵+4x⁴-16x³+64x²+16x-64=0











































Solve the polynomial and show your work.
x^7-4x^6-x^5+4x^4-16x^3+64x^2+16x-64=0

Comments for Solve the polynomial x⁷-4x⁶-x⁵+4x⁴-16x³+64x²+16x-64=0

Click here to add your own comments

Feb 24, 2011
Solve an Equation containing a 7 degree Polynomial
by: Staff


The question:

Solve the polynomial and show your work.

x^7-4x^6-x^5+4x^4-16x^3+64x^2+16x-64=0


The answer:

This equation can be factored to find the 7 solutions. However, the process is long.

I am going to show you how to factor the equation, but I’m leaving the details of each step to you.


Part I: factor (x^7-4x^6-x^5+4x^4-16x^3+64x^2+16x-64)

Step 1: divide the entire expression (x^7-4x^6-x^5+4x^4-16x^3+64x^2+16x-64) on the left side of the equation by (x + 2)

When you divide, you will see that the factors are:

(x + 2)(x^6 – 6x^5 + 11x^4 – 18x^3 + 20x^2 + 24x – 32) = 0


Step 2: divide the expression (x^6 – 6x^5 + 11x^4 – 18x^3 + 20x^2 + 24x – 32) on the left side of the equation by (x^2 - 1)

When you divide, you will see that the factors are now:

(x + 2)(x^2 – 1)(x^4 – 6x^3 + 12x^2 – 24x + 32) = 0


Step 3: divide the expression (x^4 – 6x^3 + 12x^2 – 24x + 32) on the left side of the equation by (x^2 + 4)

When you divide, you will see that the factors are now:

(x + 2)(x^2 – 1)(x^2 + 4)(x^2 – 6x + 8) = 0



Step 4: factor the expression (x^2 – 6x + 8) on the left side of the equation

When you divide, you will see that the factors are now:

(x + 2)(x^2 – 1) (x^2 + 4)(x – 2)(x – 4) = 0




Step 5: factor the expression (x^2 – 1) on the left side of the equation

When you divide, you will see that the factors are now:

(x + 2)(x + 1)(x – 1) (x^2 + 4)(x – 2)(x – 4) = 0



Part II: solve for the roots using (x + 2)(x + 1)(x – 1) (x^2 + 4)(x – 2)(x – 4) = 0

1st root:

x + 2 = 0

>>>>> x = -2


2nd root:

x + 1 = 0

>>>>> x = -1


3rd root:

x - 1 = 0

>>>>> x = +1


4th & 5th roots (imaginary roots):

x^2 + 4= 0

x^2 = -4


>>>>> x = -2i

>>>>> x = +2i


6th root:

x – 2 = 0

>>>>> x = +2


7th root:

x – 4 = 0

>>>>> x = +4


The final answer: the seven roots are: x = -2, -1, +1, -2i, +2i, +2, +4



Thanks for writing.


Staff
www.solving-math-problems.com


Click here to add your own comments

Join in and write your own page! It's easy to do. How? Simply click here to return to Math Questions & Comments - 01.



Copyright © 2008-2015. All rights reserved. Solving-Math-Problems.com