IGNOU MTE 4 2018 - Find the polynomial over R of least degree which has i − 3 and

Question

Find the polynomial over R of least degree which has i − 3 and

sqrt(7) + 5i as its roots.

02 Feb 2018

Answer :

Word Count : 354

If a polynomial has real coefficients and a complex root, its complex conjugate must also be a root. Similarly, if it has a root involving a square root that is not rational, its conjugate with respect to that radical must also be a root. The given roots are: 1. $i - 3$ → its conjugate is $-i - 3$ 2. $\sqrt{7} + 5i$ → its conjugate is $\sqrt{7} - 5i$ Thus, the roots of the polynomial over $\mathbb{R}$ are: $$ i - 3, \; -i - 3, \; \sqrt{7} + 5i, \; \sqrt{7} - 5i $$ Construct factors corresponding to each pair of conjugate roots: For $i - 3$ and $-i - 3$: $$ (x - (i-3))(x - (-i-3)) = (x + 3 - i)(x + 3 + i) $$ $$ = (x + 3)^2 - i^2 = (x + 3)^2 +

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