# 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

 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