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QUESTION

# What is the polynomial function of lowest degree with lead coefficient 1 and roots 1 and 1+i?

x^3-3x^2+4x-2,

We know that the complex roots always occur in conjugate pairs.

One complex root is 1+i, so there must be its conjugate, i.e., 1-i as the other root.

Hence, there are 3" roots :"1,1+i, 1-i.

Therefore, the poly. of the least degree must be a cubic having 3" zeroes, "1, 1+i, and, 1-i.

Since the lead-co-eff. is 1, the cubic poly. p(x) must read :

 p(x)=(x-1)(x-(1+i))(x-(1-i))

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

=(x-1){(x-1)^2-i^2}

=(x-1){(x-1)^2+1}

=(x-1)^3+(x-1)

=(x^3-1-3x^2+3x)+(x-1)

=x^3-3x^2+4x-2,