Question

obtain all other zeroes of x⁴-7x²+12 if it is given that two of its zeroes are ±√3​

Answer 1

The Fundamental Theorem of Algebra tells us that this polynomial of degree 4 has 4 roots.

The Rational Roots Method tells us that if this polynomial has rational roots, they will be

plus or minus 1, 2, 3, 4, 6, or 12

Substituting these values into the polynomial one at a time leads to

x = 2 and x = -2 as roots, therefore

(x -2) * (x +2) divides (x⁴-7x²+12), then

( x⁴-7x²+12 ) / (x^2 -4) = x^2 -3

solve x^2 -3 = 0

x = square root(3), x = -square root(3)

Therefore, the 4 roots are

x = 2, x = -2, x = square root(3), and x = -square root(3)