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COLLEGE ALGEBRA P.5 – Rational Expressions P.6 – Complex Numbers P5 – Rational Expressions A rational expression is a fraction in which the numerator and denominator are polynomials. 3 𝑥+1 𝑥 2 − 4𝑥 − 21 𝑥2 − 9 P5 – Rational Expressions The domain of a rational expression is the set of all real numbers that can be used as replacements for the variable. Any variable that causes division by zero is excluded from the domain of the rational expression. 𝑥+3 𝑥 2 − 5𝑥 What values can x not be? P5 – Rational Expressions What value of x must be excluded from the domain of? 𝑥+2 𝑥+1 P5 – Rational Expressions Properties of Rational Expressions: The following rules only work if Q and S do NOT equal 0. Equality: 𝑃 𝑄 = 𝑅 𝑆 Equivalent Expressions 𝑃 𝑄 = 𝑃𝑅 ,𝑅 𝑄𝑅 Sign 𝑃 − 𝑄 = 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑃𝑆 = 𝑄𝑅 −𝑃 𝑄 ≠0 = 𝑃 −𝑄 P5 – Simplify a Rational Expression To simplify a rational expression, factor the numerator and denominator. Then use the equivalent expressions property to eliminate factors common to both the numerator and denominator. A rational expression is simplified when 1 is the only common factor of both the numerator and the denominator. P5 – Simplify a Rational Expression Simplify: 3𝑥 2 − 20𝑥 − 7 2𝑥 2 − 11𝑥 − 21 P5 – Operations on Rational Expressions Operations on Rational Expressions The following rules only work if Q and S do NOT equal 0. Addition 𝑃 𝑄 + 𝑅 𝑄 𝑅 𝑄 𝑃+𝑅 𝑄 = Subtraction 𝑃 𝑄 − Multiplication 𝑃 𝑄 ∙ = 𝑃∙𝑅 𝑄∙𝑆 Division 𝑃 𝑄 ÷ = 𝑅 𝑆 𝑃 𝑄 𝑅 𝑆 = 𝑃−𝑅 𝑄 𝑆 𝑅 ∙ = 𝑃𝑆 𝑄𝑅 Where 𝑅 ≠ 0 P5 – Operations on Rational Expressions Multiply: 𝑥2 − 4 𝑥 2 − 11𝑥 + 28 ∙ 2 2 𝑥 + 2𝑥 − 8 𝑥 − 5𝑥 − 14 P5 – Operations on Rational Expressions Divide: 𝑥 2 + 6𝑥 + 9 𝑥 2 + 7𝑥 + 12 ∙ 3 3 𝑥 + 27 𝑥 − 3𝑥 2 + 9𝑥 P5 – Operations on Rational Expressions Addition of rational expressions with a common denominator is accomplished by writing the sum of the numerators over the common denominator. 5𝑥 𝑥 5𝑥 + 𝑥 6𝑥 𝑥 + = = = 18 18 18 18 3 If the rational expressions do not have a common denominator find the LCD: 1. Factor each denominator completely and express repeated factors using exponential notation. 2. Identify the largest power of each factor in any single factorization. The LCD is the product of each factor raised to its largest power. P5 – Operations on Rational Expressions Add: 2𝑥 + 1 𝑥 + 2 + 𝑥−3 𝑥+5 P5 – Operations on Rational Expressions Subtract: 39𝑥 + 36 23𝑥 − 16 − 2 2 𝑥 − 3𝑥 − 10 𝑥 − 7𝑥 + 10 P5 – Operations on Rational Expressions Use the Order of Operations: 𝑥 + 3 𝑥 + 4 𝑥 2 + 5𝑥 + 4 − ÷ 2 𝑥 − 2 𝑥 − 1 𝑥 + 4𝑥 − 5 P5 – Complex Fractions A complex fraction is a fraction whose numerator or denominator contains one or more fractions. Simplify complex fractions using one of the following… 1. Multiply by 1 in the form of the LCD. 1. 2. Determine the LCD of all fractions in the complex fraction. Multiply both the numerator and the denominator of the complex fraction by the LCD. 2. Multiply the numerator by the reciprocal of the denominator. 1. Simplify the numerator to a single fraction and the denominator by a single fraction. 2. Using the definition for dividing fractions, multiply the numerator by the reciprocal of the denominator. 3. If possible, simplify the resulting rational expressions. P5 – Complex Fractions Simplify: 2 1 + 𝑥−2 𝑥 3𝑥 2 − 𝑥−5 𝑥−5 P5 – Complex Fractions Simplify: 2𝑥 4− 𝑥−2 2− 𝑥 P5 – Complex Fractions Simplify: 𝑐 −1 𝑎−1 + 𝑏 −1 P5 – Complex Fractions The average speed for a round trip is given by the complex fraction: 2 1 1 + 𝑣1 𝑣2 where v1 is the average speed on the way to your destination and v2 is the average speed on your return trip. Find the average speed for a round trip of v1 = 50 mph and v2 = 40 mph. P6 – Complex Numbers Definition of i The imaginary unit, designated by the letter i is the number such that i2 = -1. The principle square root of a negative number is defined in terms of i. If a is a positive real number, then −𝑎 = 𝑖 𝑎 The number 𝑖 𝑎 is called an imaginary number. −36 = 𝑖 36 = 6𝑖 −18 = 𝑖 18 = 3𝑖 2 P6 – Complex Numbers A complex number is a number of the form a-bi, where a and b are real numbers and 𝑖 = −1. The number a is the real part of the a + bi, and b is the imaginary part. - 3 + 5i Real Part _____; Imaginary Part______ 2 - 6i Real Part _____; Imaginary Part______ 5 Real Part _____; Imaginary Part______ 7i Real Part _____; Imaginary Part______ P6 – Complex Numbers Writing a complex number in standard form a – bi. 7 + −45 P6 – Complex Numbers Addition and Subtraction of Complex Numbers: Basically add/subtract the real number parts and the imaginary number parts. 7 − 2𝑖 + (−2 + 4𝑖) P6 – Complex Numbers −9 + 4𝑖 − (2 − 6𝑖) P6 – Complex Numbers Multiply −6 ∙ −24 P6 – Complex Numbers Multiply Complex Numbers… Memorize this… 𝑖 2 = −1 3𝑖(2 − 5𝑖) P6 – Complex Numbers 2 Memorize this… 𝑖 = −1 3 − 4𝑖 2 + 5𝑖 P6 – Complex3 Numbers Recall that the number 2 is not in simplest form because there 3 𝑖 is a radical expression in the denominator. Similarly is not in simplest form because 𝑖 = −1. 3 𝑖 3𝑖 3𝑖 ∙ = 2= = −3𝑖 𝑖 𝑖 𝑖 −1 P6 – Complex Numbers Simplify 3 − 6𝑖 2𝑖 P6 – Complex Numbers Recall to simplify this; 2+ 3 , 5+ 3 we would multiply the numerator and denominator by the conjugate of 5 + 3, which is 5 − 3. What happens when we multiply a complex number by its conjugate? 2 + 5𝑖 2 − 5𝑖 P6 – Complex Numbers Divide the Complex Numbers 16 − 11𝑖 5 + 2𝑖 P6 – Complex Numbers Powers of i: 𝑖1 = 1 𝑖 2 = −1 𝑖 3 = −𝑖 𝑖4 = 1 𝑖5 = 𝑖 𝑖 6 = −1 𝑖 7 = −𝑖 𝑖8 = 1 We can find all values of i to powers by dividing the power by 4. The remainder that is left will help us evaluate the value of i. 𝑖153 So 153÷4 = 38 remainder 1; therefore, 𝑖1 = 1 Homework • Continue finding news articles for quarter project. • Chapter P Review Exercises • 103 – 120 ALL