Question

given that x=root 2 is a factor of polynomial P(x)=6x3+root2x2-10x-4root2,find all the zeroes of p(x).

Answer 1

x = √2 is a zero of p(x)
Then (x - √2) is a factor
p(x) = 6x^3 + √2x^2 - 10x -4√2
Other factors are
6x^3 + √2x^2 - 10x -4√2 / x - √2
= 6x^2 + 7√2x + 4
= 6x^2 + 3√2x+ 4√2x + 4
= 3√2x(√2x + 1) + 4(√2x + 1)
= (3√2x +4) (√2x+1)

6x^3 + √2x^2 - 10x -4√2 = (3√2x +4) (√2x+1)(x-√2)

3√2x + 4 = 0
x = -2√2/3

√2x+1 = 0
x = -√2/2