Find all zeros of the polynomial 2x4 + 7x3 − 19x2 − 14x + 30, if two of its zeros are `sqrt2` and `-sqrt2`.
Solution
We know that if x = a is a zero of a polynomial, then x - a is a factor of f(x).
Since, `sqrt2` and `-sqrt2` are zeros of f(x).
Therefore
`(x+sqrt2)(x-sqrt2)=x^2-(sqrt2)^2`
= x2 - 2
x2 - 2 is a factor of f(x). Now, we divide 2x4 + 7x3 − 19x2 − 14x + 30 by g(x) = x2 - 2 to find the zero of f(x).
By using division algorithm we have
f(x) = g(x) x q(x) - r(x)
2x4 + 7x3 − 19x2 − 14x + 30 = (x2 - 2)(2x2 + 7x - 15) + 0
2x4 + 7x3 − 19x2 − 14x + 30 `=(x+sqrt2)(x-sqrt2)(2x^2+10x-3x-15)`
2x4 + 7x3 − 19x2 − 14x + 30 `=(x+sqrt2)(x-sqrt2)[2x(x+5)-3(x+5)]`
2x4 + 7x3 − 19x2 − 14x + 30 `=(x+sqrt2)(x-sqrt2)(2x-3)(x+5)`
Hence, the zeros of the given polynomial are `-sqrt2`, `+sqrt2`, `(+3)/2`, -5.