Question

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

Answer 1

Answer: The zeroes of the equation are x=1, 3/2, 5i, -5i.

Step-by-step explanation:

Given equation $2x^4-5x^3+53x^2-125x+75=0$

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

$2(1)^4-5(1)^3+53(1)^2-125(1)+75$ 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    0

So the other factored polynomial, we get

$2x^3-3x^2+50x-75$

Factor it by grouping

$(2x^3-3x^2)+(50x-75)$

$x^2(2x-3) +25(2x-3)$

$(2x-3)(x^2+25).$

Setting each of the factors equal to 0, we get

2x-3=0

2x=3

x= 3/2.

$x^2+25 =0.$

$x^2 = -25.$

Taking square root on both sides, we get

x = ±5i

Therefore, the zeroes of the equation are x=1, 3/2, 5i, -5i.

Answer 2

$- 5i \: \: \: 5i \: \: \: \: 1 \: \: \: \: 1.5$