Question

Solve the polynomial function. Find all of the zeros; there are five zeros. Here is the equation:

X^5 - 8x^4 + 28x^3 - 56x^2 +64x - 32
Use synthetic division/quadratic equation :) Solve the polynomial function. Find all of the zeros; there are five zeros. Here is the equation:

X^5 - 8x^4 + 28x^3 - 56x^2 +64x - 32
Use synthetic division/quadratic equation :) @Mathematics

Answer 1

using synthetic division I get x=2 as one root. so expression simplifies as follows:\[x^5-8x^4+28x^3-56x^2+64x-32=(x-2)(x^4-6x^3+16x^2-24x+16)\]then split 2nd expression into product of two quadratics:\[x^4-6x^3+16x^2-24x+16=(x^2+ax+b)(x^2+cx+d)\]\[=x^4+(a+c)x^3+(d+ac+b)x^2+(ad+bc)x+bd\]this gives 4 equations with 4 unknowns:\[a+c=-6\]\[d+ac+b=16\]\[ad+bc=-24\]\[bd=16\]I then tried various combination of "b" and "d" to get "bd=16" and then checked which ones gave reasonable solutions to the other 3 equations:\[b=1, d=16\]\[b=2, d=8\]\[b=4, d=4\]only the last combination gave reasonable solutions, so ended up with:\[a=-2, b=4, c=-4, d=4\]plugged these back into equation to get:\[(x-2)(x^2-2x+4)(x^2-4x+4)\]You should be able to solve the rest from here...