Question

3. Find the four roots of the equation z^4 +4 =0 and use them to factor z^4+ 4
into quadratic factors with real coefficients.​

Answer 1

Answer:

$z = - 1 + i \\ z = 1 - i \\ z = - 1 + i \\ z = 1 - i$

Step-by-step explanation:

${z}^{4} + 4 = 0 \\ \\ { ({z}^{2} })^{2} - {(2i)}^{2} = 0 \\ ( {z}^{2} - 2i)( {z}^{2} + 2i) = 0 \\ {z}^{2} - 2i = 0 \: \: \: \: \: \: \: and \: \: \: \: \: \: \: {z}^{2} + 2i = 0 \\ {z}^{2} = 2i \: \: \: \: \: \: \: \: {z}^{2} = - 2i \\ \\ z = \sqrt{2} \sqrt{i} \: \: \: \: \: \: \: \: \: \: or \: \: \: \: \: \: \: z = \sqrt{2} \sqrt{ - i} \\ \\ z = + \sqrt{2} ( \frac{1}{ \sqrt{2} } + i \frac{1}{ \sqrt{2} } ) \: \: \: \: \: \: \: \: \: \: \: - \sqrt{2} ( \frac{1}{ \sqrt{2} } + i \frac{1}{ \sqrt{2} } )\: \: \: \: \: \: \: \: \: \: \: - \sqrt{2} ( \frac{1}{ \sqrt{2} } + i \frac{1}{ \sqrt{2} } ) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ - \sqrt{2} ( \frac{1}{ \sqrt{2} } - i \frac{1}{ \sqrt{2} } ) \: \: \: \: \: \: \: \: \: \\ \\ z = - 1 + i \: \: \: \: \: z = 1 - i \: \: \: \: \: \: \: \: or \: \: \: \: \: \: \: \: - 1 + i \: \: \: \: \: \: \: z = 1 - i$