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 Video
\(\textbf{2)}\) Simplify \( (3+i)(3-i) \)
\(\textbf{3)}\) Simplify \( (4+3i)(5-6i) \)
\(\textbf{4)}\) Simplify \( (5-2i)(7-i) \)
\(\textbf{5)}\) Simplify \( i(3+4i) \)
\(\textbf{6)}\) Simplify \( i(2-5i) \)
\(\textbf{7)}\) Simplify \( (5+3i)(5-3i) \)
\(\textbf{8)}\) Simplify \( 3(2+3i) \)
Challenge Questions
\(\textbf{9)}\) Simplify \( (a+bi)(a-bi) \)
\(\textbf{10)}\) What is \( (1+\sqrt{-1})(1-\sqrt{-1}) ? \)
\(\textbf{11)}\) Simplify \( (5i-2i^7)(7-i^3) \)
\(\textbf{12)}\) Simplify \( (3i^5+4i^7)(6i-2i^6) \)
See Related Pages\(\)
\(\bullet\text{ Complex Numbers Calculator (Symbolab.com)}\)
\(\)
\(\bullet\text{ Complex Number Multiplication Visualization on Graph} \)
\(\,\,\,\,\,\,\,\,\text{(Geogebra.org)}\)
\(\bullet\text{ Adding and Subtracting Polynomials}\)
\(\,\,\,\,\,\,\,\,(4d+7)−(2d−5)…\)
\(\bullet\text{ Multiplying Polynomials}\)
\(\,\,\,\,\,\,\,\,(x+2)(x^2+3x−5)…\)
\(\bullet\text{ Dividing Polynomials}\)
\(\,\,\,\,\,\,\,\,(x^3-8)÷(x-2)…\)
\(\bullet\text{ Dividing Polynomials (Synthetic Division)}\)
\(\,\,\,\,\,\,\,\,(x^3-8)÷(x-2)…\)
\(\bullet\text{ Synthetic Substitution}\)
\(\,\,\,\,\,\,\,\,f(x)=4x^4−3x^2+8x−2…\)
\(\bullet\text{ End Behavior}\)
\(\,\,\,\,\,\,\,\, \text{As } x\rightarrow \infty, \quad f(x)\rightarrow \infty \)
\(\,\,\,\,\,\,\,\, \text{As } x\rightarrow -\infty, \quad f(x)\rightarrow \infty… \)
\(\bullet\text{ Completing the Square}\)
\(\,\,\,\,\,\,\,\,x^2+10x−24=0…\)
\(\bullet\text{ Quadratic Formula and the Discriminant}\)
\(\,\,\,\,\,\,\,\,x=-b \pm \displaystyle\frac{\sqrt{b^2-4ac}}{2a}…\)
\(\bullet\text{ Multiplicity of Roots}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Rational Zero Theorem}\)
\(\,\,\,\,\,\,\,\, \pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 12…\)
\(\bullet\text{ Descartes Rule of Signs}\)
\(\,\)
\(\bullet\text{ Roots and Zeroes}\)
\(\,\,\,\,\,\,\,\,\text{Solve for }x. 3x^2+4x=0…\)
\(\bullet\text{ Linear Factored Form}\)
\(\,\,\,\,\,\,\,\,f(x)=(x+4)(x+1)(x−3)…\)
\(\bullet\text{ Polynomial Inequalities}\)
\(\,\,\,\,\,\,\,\,x^3-4x^2-4x+16 \gt 0…\)
\(\bullet\text{ Andymath Homepage}\)
About Andymath.com
Andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. If you have any requests for additional content, please contact Andy at tutoring@andymath.com. He will promptly add the content.
Topics cover Elementary Math, Middle School, Algebra, Geometry, Algebra 2/Pre-calculus/Trig, Calculus and Probability/Statistics. In the future, I hope to add Physics and Linear Algebra content.
Visit me on Youtube, Tiktok, Instagram and Facebook. Andymath content has a unique approach to presenting mathematics. The clear explanations, strong visuals mixed with dry humor regularly get millions of views. We are open to collaborations of all types, please contact Andy at tutoring@andymath.com for all enquiries. To offer financial support, visit my Patreon page. Let’s help students understand the math way of thinking!
Thank you for visiting. How exciting!