Multiplying Complex Numbers

When multiplying complex numbers, you distribute as you normally would when multiplying poloynomials. The major difference is all \(i^2\) need to be replaced with \(-1\) since \(i^2=-1\).

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

\(\textbf{1)}\)Simplify \( (5+i)(5-i) \)
Show Answer
The answer is \( 26 \)
Show Work
Show Video

\(\textbf{2)}\) Simplify \( (3+i)(3-i) \)
Show Answer
The answer is \( 10 \)

\(\textbf{3)}\) Simplify \( (4+3i)(5-6i) \)
Show Answer
The answer is \( 38-9i \)

\(\textbf{4)}\) Simplify \( (5-2i)(7-i) \)
Show Answer
The answer is \( 33-19i \)
Show Work

\(\,\,\,\,\,\,(5-2i)(7-i)\)
\(\,\,\,\,\,\,35-5i-14i+2i^2\)
\(\,\,\,\,\,\,35-19i+2(-1)\)
\(\,\,\,\,\,\,35-19i-2\)
\(\,\,\,\,\,\,33-19i\)
\(\,\,\,\,\,\,\)The answer is \( 33-19i \)

\(\textbf{5)}\) Simplify \( i(3+4i) \)
Show Answer
The answer is \( -4+3i \)
Show Work

\(\,\,\,\,\,\,i(3+4i)\)
\(\,\,\,\,\,\,3i+4i^2\)
\(\,\,\,\,\,\,3i+4(-1)\)
\(\,\,\,\,\,\,3i-4\)
\(\,\,\,\,\,\,\)The answer is \( -4+3i \)

\(\textbf{6)}\) Simplify \( i(2-5i) \)
Show Answer
The answer is \( 5+2i \)
Show Work

\(\,\,\,\,\,\,i(2-5i)\)
\(\,\,\,\,\,\,2i-5i^2\)
\(\,\,\,\,\,\,2i-5(-1)\)
\(\,\,\,\,\,\,2i+5\)
\(\,\,\,\,\,\,\)The answer is \( 5+2i \)

\(\textbf{7)}\) Simplify \( (5+3i)(5-3i) \)
Show Answer
The answer is \( 34 \)
Show Work

\(\,\,\,\,\,\,(5+3i)(5-3i)\)
\(\,\,\,\,\,\,25-15i+15i-9i^2\)
\(\,\,\,\,\,\,25-9i^2\)
\(\,\,\,\,\,\,25-9(-1)\)
\(\,\,\,\,\,\,25+9\)
\(\,\,\,\,\,\,\)The answer is \( 34 \)

\(\textbf{8)}\) Simplify \( 3(2+3i) \)
Show Answer
The answer is \( 6+9i \)
Show Work

\(\,\,\,\,\,\,3(2+3i)\)
\(\,\,\,\,\,\,6+9i\)
\(\,\,\,\,\,\,\)The answer is \( 6+9i \)

Challenge Questions

\(\textbf{9)}\) Simplify \( (a+bi)(a-bi) \)
Show Answer
The answer is \( a^2+b^2 \)

\(\textbf{10)}\) What is \( (1+\sqrt{-1})(1-\sqrt{-1}) ? \)
Show Answer
The answer is \( 2 \)

\(\textbf{11)}\) Simplify \( (5i-2i^7)(7-i^3) \)
Show Answer
The answer is \( -7+49i \)
Show Work

\( (5i-2i^7)(7-i^3) \)
\( 35i-5i^4-14i^7+2i^{10}\)
\( 35i-5i^4-14i^3+2i^{2}\)
\( 35i-5(1)-14(-i)+2(-1)\)
\( 35i-5+14i-2\)
\( -7+49i \)

\(\textbf{12)}\) Simplify \( (3i^5+4i^7)(6i-2i^6) \)
Show Answer
The answer is \( 6-2i \)

See Related Pages\(\)

\(\bullet\text{ Complex Numbers Calculator (Symbolab.com)}\)
\(\)

\(\bullet\text{ Complex Number Multiplication Visualization on Graph} \)
\(\,\,\,\,\,\,\,\,\text{(Geogebra.org)}\)

\(\bullet\text{ Adding and Subtracting Polynomials}\)
\(\,\,\,\,\,\,\,\,(4d+7)−(2d−5)…\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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