College Algebra Textbook Supplement -Acosta & Karwowski - Fall 2013 - College Algebra with Current Interesting Applications and Facts by Acosta

# College Algebra Textbook Supplement -Acosta & Karwowski - Fall 2013

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College Algebra with Current Interesting Applications and Facts by Acosta & Karwowski, Kendall Hunt, 2012 Textbook Supplement Contents A. Complex Numbers 2 B. Distance Formula, Midpoint Formula, and Circles 9 C. Synthetic Division 23 Topic Expansion from Content Already in Textbook D. More on Rational Functions 33 E. Non-Linear Systems of Equations 47 F. Other Types of Equations 61 G. Solving Polynomial and Rational Inequalities 71
2 Section A: Complex Numbers Suppose we wish to solve the equation x 2 = 25. Since we know that the square of a real number is either zero or positive, there is no real number that would satisfy this equation. To solve this problem, mathematicians created a number system that is based upon a new number: the imaginary unit, commonly referred to as " i ." Little Facts : It is said that the name "imaginary number" was originally coined by René Descartes in the seventeenth century as a derogatory term, because at that time such numbers were regarded by some as fictitious or useless. Swiss mathematician Leonhard Euler introduced the letter "i" to represent the square roots of negatives in 1777. In modern times these numbers have essential, concrete applications in math, physics, electrical engineering, and many other scientific and related areas. Sources: The Imaginary Unit The imaginary unit , i , is defined with the following properties i = √−1 and i 2 = √−1 √−1 = 1 We can use the imaginary unit to rewrite the square root of a negative number. Rewriting the Expression √− If a > 0 , then √− = i √ Let us rewrite √−25 as a product of a real number and i : √−25 = (−1)(25) = i √25 = 5 i We can check the answer by squaring 5 i : (5 i ) 2 = 5 2 i 2 = (25)( 1) = 25 Example 1 Rewrite √−28 in terms of i , and simplify if possible. Solution √−28 = i √28 = i √4 7 = 2 i √7 Note: It is also acceptable to write an expression like 2 i √7 as 2 √7 i, but we must be sure to write the " i " outside the radical symbol. To avoid being read as being under the radical, we generally write the answer with " i " in front of the radical. When mathematicians added a real number to multiples of imaginary units, the set of complex numbers was formed. Complex Number A complex number is one of the form a + bi , where a and b are real numbers. In a complex number, we call a the real part and b is the imaginary part. Two complex numbers a + bi and c + di are equal if and only if a = c and b = d . Any real number, a , can be written as a complex number as a + 0 i . In this case, b = 0. If a complex number has b 0, then we call a + bi , an imaginary number (nonreal complex
3 number) . On the other hand, if b 0 but a = 0, then a + bi = 0 + bi = bi and we call this a pure imaginary number . Some examples of complex numbers are: 2 + 5 i imaginary number, a 0 b 0 7 i pure imaginary number, a = 0 12 real number, b = 0 A complex number written in the form a + bi is said to be in standard form .