Question

Find the polynomial over R of least degree which has i−3 and √7+5i as its
roots.

Answer 1

Answer on Question #78501 – Math – Other

Question

Find the polynomial over $R$ of least degree which has $i - 3$ and $\sqrt{7} + 5i$ as its roots.

Solution

The polynomial of least degree, which has roots $x_{1} = i - 3$ and $x_{2} = \sqrt{7} + 5i$ is given by

$$P(x) = (x - x_{1})(x - x_{2})$$

But it has complex coefficients. To get the polynomial $Q(x)$ of least degree over $\mathbb{R}$ we must multiply polynomial $P(x)$ by $(x - \overline{x_1})(x - \overline{x_2})$, where $\overline{x_1} = -i - 3$, $\overline{x_2} = \sqrt{7} - 5i$. Thus we obtain

$$\begin{aligned} Q(x) &= (x - x_{1})(x - \overline{x_{1}})(x - x_{2})(x - \overline{x_{2}}) = \\ &= (x^{2} - x(x_{1} + \overline{x_{1}}) + \overline{x_{1}}x_{1})(x^{2} - x(x_{2} + \overline{x_{2}}) + \overline{x_{2}}x_{2}) = \\ &= (x^{2} + 6x + 10)(x^{2} - 2\sqrt{7}x + 32) = \\ &= 320 + 192x - 20\sqrt{7}x + 42x^{2} - 12\sqrt{7}x^{2} + 6x^{3} - 2\sqrt{7}x^{3} + x^{4} \end{aligned}$$

**Answer**: $320 + 192x - 20\sqrt{7}x + 42x^{2} - 12\sqrt{7}x^{2} + 6x^{3} - 2\sqrt{7}x^{3} + x^{4}$.

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