# Find the polynomial of degree 4 whose sum of two of the zeroes is -1 and product of these two zeroes is -2. The other two zeroes are sqrt(3) and -sqrt(3)

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A fourth degree polynomial has 2 zeroes with sum -1 and product -2. The other zeroes are `sqrt 3` and `-sqrt 3` .

Let the zeroes that are not known be A and B.

A + B = -1 and A*B = -2

Substituting A = -1 - B in A*B = -2

=> (-1 - B)*B = -2

=> B + B^2 - 2 = 0

=> B^2 + 2B - B - 2 = 0

=> B(B + 2) - 1(B + 2) = 0

=> B = 1 and B = -2

The roots are A = -2 and B = 1

This gives the polynomial as `(x - 1)(x + 2)(x - sqrt 3)(x + sqrt 3)`

=> x^4 + x^3 - 5x^2 - 3x + 6

**The polynomial is x^4 + x^3 - 5x^2 - 3x + 6**

If `a` is a root (zero) of a polynomial, then `(x-a)` is a factor of the polynomial. Since 1 is a root, (x-1) is a factor, -2 is a root then (x-(-2)) or (x+2) is a factor.

This gives the polynomial as `(x - 1)(x + 2)(x - sqrt 3)(x + sqrt 3)` pls explain this part...crucial for all questions like this...

thanks