Notes

 

\(\text{Most Important Note}\)
\(i^2=-1\)

 

 

\(\text{Other Notes}\)
\(i=\sqrt{-1}\)
\(i^2=-1\)
\(i^3=-i\)
\(i^4=1\)
\(i^5=i\)
\(i^6=-1\)
\(i^7=-i\)
\(i^8=1\)

 

 

Questions & Solutions

\(\hspace{-12pt}\small{\textbf{1)}}\)Simplify \( \displaystyle\frac{2i}{3+4i} \)

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The answer is \( \frac{6}{25}i+\frac{8}{25} \)

 

\(\hspace{-12pt}\small{\textbf{2)}}\)Simplify \( \displaystyle\frac{3-2i}{4i} \)

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The answer is \( \frac{1}{2}-\frac{3}{4}i \)

 

\(\hspace{-12pt}\small{\textbf{3)}}\)Simplify \( \displaystyle\frac{6+i}{3+2i} \)

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The answer is \( \frac{20}{13}-\frac{9}{13}i \)

 

\(\hspace{-12pt}\small{\textbf{4)}}\)Simplify \( \displaystyle\frac{3-i}{5+3i} \)

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The answer is \( \frac{6}{17}-\frac{7}{17}i \)

Show Work

\(\,\,\,\,\,\,\displaystyle\frac{3-i}{5+3i} \cdot \frac{5-3i}{5-3i}\)
\(\,\,\,\,\,\,\displaystyle\frac{15-9i-5i+3i^2}{25-15i+15i-9i^2}\)
\(\,\,\,\,\,\,\displaystyle\frac{15-14i+3i^2}{25-9i^2}\)
\(\,\,\,\,\,\,\displaystyle\frac{15-14i+3(-1)}{25-9(-1)}\)
\(\,\,\,\,\,\,\displaystyle\frac{15-14i-3}{25+9}\)
\(\,\,\,\,\,\,\displaystyle\frac{12-14i}{34}\)
\(\,\,\,\,\,\,\)The answer is \( \frac{6}{17}-\frac{7}{17}i \)

 

\(\hspace{-12pt}\small{\textbf{5)}}\)Simplify \( \displaystyle\frac{3-2i}{3-i} \)

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The answer is \( \displaystyle\frac{11}{10}-\frac{3}{10}i \)

Show Work

\( \displaystyle\frac{3-2i}{3-i} \)
\( \displaystyle\frac{3-2i}{3-i} \cdot \frac{3+i}{3+i}\)
\( \displaystyle\frac{9+3i-6i-2i^2}{9+3i-3i-i^2}\)
\( \displaystyle\frac{9+3i-6i-2(-1)}{9+3i-3i-(-1)}\)
\( \displaystyle\frac{9+3i-6i+2}{9+3i-3i+1}\)
\( \displaystyle\frac{11-3i}{10}\)
\( \displaystyle\frac{11}{10}-\frac{3}{10}i \)

 

\(\hspace{-12pt}\small{\textbf{6)}}\)Simplify \( \displaystyle\frac{i}{9-2i} \)

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The answer is \( -\frac{2}{85}+\frac{9}{85}i \)

Show Work

\( \displaystyle\frac{i}{9-2i} \)
\( \displaystyle\frac{i}{9-2i} \cdot \frac{9+2i}{9+2i}\)
\( \displaystyle\frac{i(9+2i)}{(9-2i)(9+2i)}\)
\( \displaystyle\frac{9i + 2i^2}{81 – (2i)^2}\)
\( \displaystyle\frac{9i + 2(-1)}{81 – 4(-1)}\)
\( \displaystyle\frac{9i – 2}{81 + 4}\)
\( \displaystyle\frac{-2 + 9i}{85}\)
\( \displaystyle-\frac{2}{85}+\frac{9}{85}i \)

 

Challenge Questions

\(\hspace{-12pt}\small{\textbf{7)}}\)Simplify \( \displaystyle\frac{5-i^5}{2-3i^3} \)

Show Answer

The answer is \(\frac{7}{13}-\frac{17}{13}i \)

 

See Related Pages\(\)

\(\bullet\text{ Complex Numbers Calculator (Symbolab.com)}\)
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\(\bullet\text{ Adding and Subtracting Polynomials}\)
\(\,\,\,\,\,\,\,\,(4d+7)−(2d−5)…\)

\(\bullet\text{ Multiplying Polynomials}\)
\(\,\,\,\,\,\,\,\,(x+2)(x^2+3x−5)…\)

\(\bullet\text{ Dividing Polynomials}\)
\(\,\,\,\,\,\,\,\,(x^3-8)÷(x-2)…\)

\(\bullet\text{ Dividing Polynomials (Synthetic Division)}\)
\(\,\,\,\,\,\,\,\,(x^3-8)÷(x-2)…\)

\(\bullet\text{ Synthetic Substitution}\)
\(\,\,\,\,\,\,\,\,f(x)=4x^4−3x^2+8x−2…\)

\(\bullet\text{ End Behavior}\)
\(\,\,\,\,\,\,\,\, \text{As } x\rightarrow \infty, \quad f(x)\rightarrow \infty \)
\(\,\,\,\,\,\,\,\, \text{As } x\rightarrow -\infty, \quad f(x)\rightarrow \infty… \)

\(\bullet\text{ Completing the Square}\)
\(\,\,\,\,\,\,\,\,x^2+10x−24=0…\)

\(\bullet\text{ Quadratic Formula and the Discriminant}\)
\(\,\,\,\,\,\,\,\,x=-b \pm \displaystyle\frac{\sqrt{b^2-4ac}}{2a}…\)

\(\bullet\text{ Multiplicity of Roots}\)
\(\,\,\,\,\,\,\,\,\)\(…\)

\(\bullet\text{ Rational Zero Theorem}\)
\(\,\,\,\,\,\,\,\, \pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 12…\)

\(\bullet\text{ Descartes Rule of Signs}\)
\(\,\)

\(\bullet\text{ Roots and Zeroes}\)
\(\,\,\,\,\,\,\,\,\text{Solve for }x. 3x^2+4x=0…\)

\(\bullet\text{ Linear Factored Form}\)
\(\,\,\,\,\,\,\,\,f(x)=(x+4)(x+1)(x−3)…\)

\(\bullet\text{ Polynomial Inequalities}\)
\(\,\,\,\,\,\,\,\,x^3-4x^2-4x+16 \gt 0…\)

\(\bullet\text{ Andymath Homepage}\)