Question

What is the polynomial of the smallest degree with integer coefficients and with zero's 3,1+i,-1?

Answer 1

Chapter - Polynomials

Question. What is the polynomial of the smallest degree with integer coefficients and with zeroes 3, 1 + i, 1 - i ?

Solution.

We know that, if a, b, c be the zeroes of a polynomial, then the polynomial be

p(x) = (x - a) (x - b) (x - c)

Using the above formula, we get the required polynomial

p(x) = (x - 3) {x - (1 + i)} {x - (1 - i)}

= (x - 3) {x² - (1 + i + 1 - i) x + (1^2 - i^2)}

= (x - 3) {x² - 2x + (1 + 1)}

= (x - 3) (x² - 2x + 2)

= x³ - 2x² + 2x - 3x² + 6x - 6

= x³ - 5x² + 8x - 6

This is the required polynomial.

Answer. Required polynomial = x³ - 5x² + 8x - 6.

Answer 2

Answer :

Root = 3, 1+i, -1

Therefore, We know that no real roots always exist in pairs.

So, one more root will be  1 -i

Therefore, sum of root = 4

And product of the root = 3(-1) (1+1) (1-i)

                                        = -3 (1- $i^{2}$)

                                        = -3 (2)

                                        = -6

Hope this will help you.