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Question

Answer

$3 (1 + 2 i)^{4} + 12 (1 + 2 i)^{3} + 39 (1 + 2 i)^{2} = 60 i - 270$

Explanation

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$3 (1 + 2 i)^{4} + 12 (1 + 2 i)^{3} + 39 (1 + 2 i)^{2} \overset{◯}{1} \overset{◯}{2} \overset{◯}{3} \overset{◯}{4} 3 (16 i^{4} + 32 i^{3} + 24 i^{2} + 8 i + 1) + 12 (1 + 6 i + 12 i^{2} + 8 i^{3}) + 39 (1 + 4 i + 4 i^{2}) \overset{◯}{5} \overset{◯}{6} \overset{◯}{7} \overset{◯}{8} 3 (16 - 32 i - 24 + 8 i + 1) + 12 (1 + 6 i - 12 - 8 i) + 39 (1 + 4 i - 4) \overset{◯}{9} 3 (- 24 i - 7) + 12 (- 2 i - 11) + 39 (4 i - 3) \overset{◯}{10} - 72 i - 21 - 24 i - 132 + 156 i - 117 \overset{◯}{11} - 96 i - 153 + 156 i - 117 \overset{◯}{12} 60 i - 270$
①	$(1 + 2 i)^{4} = (1 + 2 i)^{2} \cdot (1 + 2 i)^{2}$
②	Find $(1 + 2 i)^{2}$ using formula. $(A + B)^{2} = A^{2} + 2 \cdot A \cdot B + B^{2}$ where $A = 1$ and $B = 2 i$ . $(1 + 2 i)^{2} = 1^{2} + 2 \cdot 1 \cdot 2 i + (2 i)^{2} = 1 + 4 i + 4 i^{2}$
③	Multiply each term of $(1 + 4 i + 4 i^{2})$ by each term in $(1 + 4 i + 4 i^{2})$ . $(1 + 4 i + 4 i^{2}) \cdot (1 + 4 i + 4 i^{2}) = 1 + 4 i + 4 i^{2} + 4 i + 16 i^{2} + 16 i^{3} + 4 i^{2} + 16 i^{3} + 16 i^{4}$
④	Combine like terms: $1 + 4 i + 4 i^{2} + 4 i + 16 i^{2} + 16 i^{3} + 4 i^{2} + 16 i^{3} + 16 i^{4} = = 16 i^{4} + 32 i^{3} + 24 i^{2} + 8 i + 1$ Find $(1 + 2 i)^{3}$ using formula $(A + B) = A^{3} + 3 A^{2} B + 3 A B^{2} + B^{3}$ where $A = 1$ and $B = 2 i$ . $(1 + 2 i)^{3} = 1^{3} + 3 \cdot 1^{2} \cdot 2 i + 3 \cdot 1 \cdot (2 i)^{2} + (2 i)^{3} = 1 + 6 i + 12 i^{2} + 8 i^{3}$ Find $(1 + 2 i)^{2}$ using formula. $(A + B)^{2} = A^{2} + 2 \cdot A \cdot B + B^{2}$ where $A = 1$ and $B = 2 i$ . $(1 + 2 i)^{2} = 1^{2} + 2 \cdot 1 \cdot 2 i + (2 i)^{2} = 1 + 4 i + 4 i^{2}$
⑤	$16 i^{4} = 16 \cdot i^{2} \cdot i^{2} = 16 \cdot (- 1) \cdot (- 1) = 16$
⑥	$32 i^{3} = 32 \cdot i^{2} \cdot i = 32 \cdot (- 1) \cdot i = - 32 \cdot i$
⑦	$24 i^{2} = 24 \cdot (- 1) = - 24$ $12 i^{2} = 12 \cdot (- 1) = - 12$
⑧	$8 i^{3} = 8 \cdot i^{2} \cdot i = 8 \cdot (- 1) \cdot i = - 8 \cdot i$ $4 i^{2} = 4 \cdot (- 1) = - 4$
⑨	Combine like terms: $16 - 32 i - 24 + 8 i + 1 = - 24 i - 7$ Combine like terms: $1 + 6 i - 12 - 8 i = - 2 i - 11$ Combine like terms: $1 + 4 i - 4 = 4 i - 3$
⑩	Multiply $3$ by $(- 24 i - 7)$ $3 \cdot (- 24 i - 7) = - 72 i - 21$ Multiply $12$ by $(- 2 i - 11)$ $12 \cdot (- 2 i - 11) = - 24 i - 132$ Multiply $39$ by $(4 i - 3)$ $39 \cdot (4 i - 3) = 156 i - 117$
⑪	Combine like terms: $- 72 i - 21 - 24 i - 132 = - 96 i - 153$
⑫	Combine like terms: $- 96 i - 153 + 156 i - 117 = 60 i - 270$

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