Question

If √2 and√3 are two zeroes of x^4+-5x^2+6. Find other two zeroes

Answer 1

Two zeros of x⁴ - 5x² + 6 are √2 and √3 .
we can resolve given Polynomial.
x⁴ - 5x² + 6
= x⁴ - √2x³ + √2x³ - 2x² - 3x² + 3√2x - 3√2x + 6
= x³(x - √2) + √2x²(x - √2) - 3x(x - √2) - 3√2(x - √2)
= (x -√2){x³ + √2x² - 3x - 3√2}
= (x - √2){x³ -√3x² + √3x² + √2x² - 3x - √6x + √6x - 3√2}
= (x - √2){x²(x - √3) + (√3+ √2)x(x -√3) + √6(x - √3)}
= (x - √2)(x - √3)(x² + √3x + √2x + √6)

Hence, next two zeros in (x² + √3x + √2x + √6)
Solve this polynomial,
x² + √3x + √2x + √6
= x(x + √3) + √2(x + √3)
= (x + √3)(x + √2)

Hence, other two zeros are - √3 and -√2