Find all the zeroes of the equation 2x^4-5x^3+53x^2-125x+75=0

Answer: The zeroes of the equation are x=1, 3/2, 5i, -5i.
Step-by-step explanation: Given equation [tex]2x^4-5x^3+53x^2-125x+75=0[/tex]Applying rational roots theorem.The constant term is 75 and leading coefficient is 2.Factors of 75 are 1,3,5,15. and factors of 2 are 1, and 2.Therefore, possible rational roots would be ±1,3,5,15,1/2, 3/2, 5/2 and 15/2.Let us check first x=1 if it is a root or not.Plugging x=1 in given equation, we get [tex]2(1)^4-5(1)^3+53(1)^2-125(1)+75[/tex] would give us 0.Therefore, first root would be x=1 so the first factor would be x-1.Dividing given polynomial using syntactic division ________________________1        |      2   -5   53   -125    75                      2    -3    +50    -75_______________________               2   -3   +50   -75    0So the other factored polynomial, we get [tex]2x^3-3x^2+50x-75[/tex]Factor it by grouping [tex](2x^3-3x^2)+(50x-75)[/tex][tex]x^2(2x-3) +25(2x-3)[/tex][tex](2x-3)(x^2+25).[/tex]Setting each of the factors equal to 0, we get 2x-3=0 2x=3 x= 3/2.[tex]x^2+25 =0.[/tex][tex]x^2 = -25.[/tex]Taking square root on both sides, we get x = ±5iTherefore, the zeroes of the equation are x=1, 3/2, 5i, -5i.