List all possible or potential rational zeros for the polynomial below find all real zeros of the polynomial below and factor completely over the real numbers.

Question

f(x) x^4 + 10x^3 + 12x^2 - 58x + 35
Someone anyone?????

Answer 1

if any factors exist then, then some of ±1, ±5, ±7 must be zeros of the function
it is easy to see that x=1 works
f(1) = 1+10+12-58+35 = 0
by synthetic division,
x^4 + 10x^3 + 12x^2 - 58x + 35 = (x-1)(x^3 + 11x^2 + 23x - 35)

mmmh, looks like x=1 is also a zero of the cubic facor
another sysnthetic division left me with
x^4 + 10x^3 + 12x^2 - 58x + 35 =(x-1)(x-1)(x^2 + 12x + 35)
and that quadratic obviously factors again, so we have

x^4 + 10x^3 + 12x^2 - 58x + 35 =(x-1)^2 (x+5)(x+7)

Answer 2

Thank you!