Eng’g 10 PRE-CALCULUS Rational Expressions Pre-Calculus 1 Learning Outcomes After careful study of this chapter, students should be able to do the follow ing: 1. Perform calculations on Radicals by applying the properties; 2. Solve problems on complex numbers. Rational Expressions Pre-Calculus 2 A. Rational Expressions A rational expression is an expression that is the ratio of two polynomials. Example: It is just like a fraction, but with polynomials. Rational Expressions Pre-Calculus 3 A. Rational Expressions In General A rational function is the ratio of two polynomials P(x) and Q(x) like this 𝑁(𝑥) 𝑓 𝑥 = 𝐷(𝑥) Except that 𝐷(𝑥) cannot be zero since 1 = undefined. 0 Rational Expressions Pre-Calculus 4 A. Rational Expressions Rational Expressions Pre-Calculus 5 A. Rational Expressions Simplification of Rational Expressions To simplify rational expressions we must follow the following steps: 1. Factor numerator and denominator. 2. Simplify. Example 1: Simplify the following rational expression (x² + 8x + 12) / (x² - 36). We factor the numerator and the denominator, remembering that (a² - b²) = (a + b) (a - b): Rational Expressions Pre-Calculus 6 Simplification of Rational Expressions 2 𝑥 + 8𝑥 + 12 𝑥+6 𝑥+2 = 𝑥 2 − 36 𝑥−6 𝑥+6 Cancel the common factor 𝑥 + 6 𝑥 2 + 8𝑥 + 12 𝒙+𝟐 = 2 𝑥 − 36 𝒙−𝟔 Rational Expressions Pre-Calculus 7 Simplification of Rational Expressions Example 2: Simplify the following rational expression (n³ - n) / (n² - 5n - 6). We factor the numerator and the denominator, 𝑛3 − 𝑛 𝑛(𝑛 + 1)(𝑛 − 1) = 𝑛2 − 5𝑛 − 6 𝑛−6 𝑛−1 Cancel the common factor 𝑛 − 1 3 𝑛 −𝑛 𝒏(𝒏 + 𝟏) = 𝑛2 − 5𝑛 − 6 𝒏−𝟔 Rational Expressions Pre-Calculus 8 Operations on Rational Expressions Addition The easiest way to add rational expressions is to use the common denominator. Example 1: 2 3 2 𝑥+1 +3 𝑥−2 + = 𝑥−2 𝑥+1 𝑥−2 𝑥+1 Then simplify to: 2 3 2𝑥 + 2 + 3𝑥 − 6 5𝑥 − 4 + = = 2 𝑥−2 𝑥+1 𝑥−2 𝑥+1 𝑥 −𝑥−2 Rational Expressions Pre-Calculus 9 Operations on Rational Expressions Subtraction Subtracting is just like Adding: Example 2: 2 3 2 𝑥+1 −3 𝑥−2 − = 𝑥−2 𝑥+1 𝑥−2 𝑥+1 Then simplify to: 2 3 2𝑥 + 2 − (3𝑥 − 6) −𝑥 + 8 + = = 2 𝑥−2 𝑥+1 𝑥−2 𝑥+1 𝑥 −𝑥−2 Rational Expressions Pre-Calculus 10 Operations on Rational Expressions Multiplication To multiply two rational expressions, just multiply the tops and bottoms separately Example 3: 2 𝑥−2 3 2 3 = 𝑥+1 𝑥−2 𝑥+1 Then simplify to: 2 𝑥−2 Rational Expressions 3 6 = 2 𝑥+1 𝑥 −𝑥−2 Pre-Calculus 11 Operations on Rational Expressions Division To divide two rational expressions, first flip the second expression over (make it a reciprocal) and then multiply. Example 4: 2 3 2 / = 𝑥−2 𝑥+1 𝑥−2 𝑥+1 3 Then simplify to: 2 3 2𝑥 + 2 / = 𝑥−2 𝑥+1 3𝑥 − 6 Rational Expressions Pre-Calculus 12 B. Radicals A radical is an expression of 𝑛 𝑎 denoting the principal 𝑛𝑡ℎ root positive integer 𝑛 is the index, or the radical and the number 𝑎 is the The index is omitted if 𝑛 = 2. the form of 𝑎. The order, of radicand. The laws for radicals are obtained directly from the laws for exponents by means of the definition 𝑛 Rational Expressions 𝑎𝑚 = 𝑚 𝑎𝑛 Pre-Calculus 13 Properties of Radicals If 𝑛 is even, assume a, b ≥ 0. Rational Expressions Pre-Calculus 14 Simplification of Radicals It is important to reduce a radical to its simplest form through use of the following operations. 1. Removal of perfect 𝑛𝑡ℎ 𝑝𝑜𝑤𝑒𝑟𝑠 from a radicand. Any radical of order n should be simplified by removing all perfect 𝑛𝑡ℎ powers from under the radical sign using the rule Rational Expressions Pre-Calculus 15 Simplification of Radicals Examples: 2. Reduction of the index of the radical. Examples: Rational Expressions Pre-Calculus 16 Simplification of Radicals A radical is said to be in simplest form if: 1. All perfect n-th powers have been removed from the radical. 2. The index of the radical is as small as possible. 3. No fractions are present in the radicand i.e. the denominator has been rationalized. Rational Expressions Pre-Calculus 17 Simplification of Radicals Similar Radicals Radicals which, on being reduced to simplest form, have the same index and radicand. Rational Expressions Pre-Calculus 18 C. Operations on Radicals Addition and Subtraction of Radicals Before addition or subtraction of radicals it is important to reduce them to simplest form. Like radicals can then be added or subtracted in the same way as other like terms. Example: Rational Expressions Pre-Calculus 19 Operations on Radicals Multiplication of Radicals 1. To multiply two or more radicals having the same index use 𝑛 𝑛 𝑛 𝑎 𝑏 = 𝑎𝑏 Example: Rational Expressions Pre-Calculus 20 Multiplication of Radicals 1. To multiply radicals with different indices use fractional exponents and the laws of exponents. Example: Rational Expressions Pre-Calculus 21 Operations on Radicals Division of Radicals 1. To divide two radicals having the same index use 𝑛 𝑛 𝑎 𝑏 = 𝑛 𝑎 , 𝑏 and simplify. Example: Rational Expressions Pre-Calculus 22 Division of Radicals 2. To divide radicals with different indices use fractional exponents and the laws of exponents. Example: Rational Expressions Pre-Calculus 23 Rationalization In one form of fraction, the denominator is a binomial in which one term is a square root of a rational number or expression and the other term is of the same form or is rational i.e. the denominator has the form 𝑎 + 𝑏 or 𝑎 + 𝑏 . In such a case, rationalize the fraction by multiplying the numerator and denominator by the conjugate of the denominator. Rational Expressions Pre-Calculus 24 Rationalization where the conjugate of 𝑎 + 𝑏 = 𝑎 − 𝑏 and the conjugate of 𝑎+𝑏 = 𝑎− 𝑏.Example: Rational Expressions Pre-Calculus 25 D. Complex Numbers A Complex Number is a combination of a Real Number and an Imaginary Number. Either Part Can Be Zero So, a Complex Number has a real part and an imaginary part. Rational Expressions Pre-Calculus 26 Complex Numbers But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers. Rational Expressions Pre-Calculus 27 Complex Numbers Rational Expressions Pre-Calculus 28 Complex Numbers Rational Expressions Pre-Calculus 29 Complex Numbers Rational Expressions Pre-Calculus 30 Complex Numbers Rational Expressions Pre-Calculus 31 Complex Numbers Rational Expressions Pre-Calculus 32 Complex Numbers Rational Expressions Pre-Calculus 33 Complex Numbers Rational Expressions Pre-Calculus 34 Complex Numbers Rational Expressions Pre-Calculus 35 Complex Numbers Rational Expressions Pre-Calculus 36 Complex Numbers Rational Expressions Pre-Calculus 37 Complex Numbers Rational Expressions Pre-Calculus 38 Complex Numbers Rational Expressions Pre-Calculus 39 References • Feliciano and Uy, College Algebra • Dugopolski, Mark. College Algebra, 3rd ed. Addison-Wesley, 2002. • Mijares, Catalina. College Algebra. Rational Expressions Pre-Calculus 40 For queries contact Engr. ODETTE R. ALEGATO email add: [email protected] Rational Expressions Pre-Calculus 41
