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PREC11 Unit5 notes - PRE-CALCULUS 11 Unit 5 Day 1 RATIONAL...

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PRE-CALCULUS 11 Unit 5 – Day 1: RATIONAL EXPRESSIONS (Part 1) RATIONAL EXPRESSIONS A rational number is any number that can be written as a quotient of two integers, b a where the denominator cannot equal zero; a 0 . A rational expression is any expression that can be written as a quotient of two polynomials, polynomial polynomial where the denominator cannot equal zero. Some examples of rational expressions are x x 3 6 , x x x - + - 2 3 7 2 3 , and x 4 - 3 . Rational expressions are also known as algebraic fractions . Restrictions on Rational Expressions A rational expression is not defined when its denominator is equal to zero. Restrictions on rational expressions begin with "denominator 0". Any variable value that results in a zero denominator is a non-permissible value. examples: Find the restrictions on each rational expression. a) a ab 3 ab 0 ............................................... a 0 and b 0 b) x x - 2 4 9 2 3 2 x + 3 0 .......................................... x - 3 2 c) x x x x - - + 2 2 4 12 5 6 x 2 - 5 x + 6 0; ( x - 2)( x - 3) 0 ....... x 2 and x 3 d) a a - 2 2 4 4 a 2 + 4 0; a 2 - 4 ............................. no restrictions exercises: State any non-permissible values for x . a) x x x + - + 2 3 18 6 b) x x x - - 2 2 6 c) x x x - + - 2 2 5 6 9 [Answers: - 6, - 2 and 3, - 3 and 3]
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Unit 5: Day 1 notes - Rational Expressions (Part 1) Page 2 of 2 SIMPLIFYING RATIONAL EXPRESSIONS 14 21 can be simplified by dividing the numerator and denominator by the same value. The number that is being used as the divisor must be a common factor of the numerator and denominator; (2)(7) (3)(7) = 2 3 This same procedure can be used to simplify rational expressions. examples: Simplify each rational expression. a) x x - 2 4 9 2 3 b) x x x x - - + 2 2 4 12 5 6 c) a a a - - + 2 2 1 2 1 x x x - + (2 3)(2 3) 2 3 x x x x - - - 4 ( 3) ( 2)( 3) a a a a - - - (1 )(1 ) ( 1)( 1) a a a a - - - - (1 )( 1)( 1) ( 1)( 1) answers: 2 x - 3 x x - 4 2 a a - - 1 1 exercises: Simplify. a) x x x + - + 2 3 18 6 b) x x x - - 2 2 6 c) x x x - + - 2 2 5 6 9 [ Answers: x - 3, x - 1 3 , x x - - 2 3 ]
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PRE-CALCULUS 11 Unit 5 – Day 2: RATIONAL EXPRESSIONS (Part 2) RATIONAL EXPRESSIONS Rational expressions can be used to model many situations. example: The volume of a right rectangular prism in cubic centimetres is given by the polynomial x 3 + 4 x 2 - 12 x , where x is the width of the prism's base in centimetres. The length of the prism's base is ( x + 6) centimetres. a) Write a rational expression that will represent the prism's height. b) Simplify the rational expression. What does this expression represent? c) State the permissible values for x . solution a) The volume of a right rectangular prism is the product of its base area and its height. The base area is the product of the base's width and length. The prism's height must be the result of dividing its volume by the product of the base's width and length. The prism's height is x x x x x + - + 3 2 4 12 ( 6) b) To simplify the numerator must be factored. x x x x x + - + 2 ( 4 12) ( 6) x x x x x - + ( 6)( 2) ( 6) Divide the numerator and denominator by the common factors. x - 2 x - 2 represents the prism's height in centimetres.
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