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6.1 The Fundamental Property of Rational Expressions

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Rational Expression A rational expression is an expression of the form where P and Q are polynomials, with Q ≠ 0. Examples of rational expressions The Fundamental Property of Rational Expressions The quotient of two integers (with the denominator not 0), such as or is called a rational number. In the same way, the quotient of two polynomials with the denominator not equal to 0 is called a rational expression. Slide 6.1-3

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Objective 1 Find the numerical value of a rational expression. Slide 6.1-4

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Find the value of the rational expression, when x = 3. Solution: Slide 6.1-5 Evaluating Rational Expressions CLASSROOM EXAMPLE 1

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Objective 2 Find the values of the variable for which a rational expression is undefined. Slide 6.1-6

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Find the values of the variable for which a rational expression is undefined. In the definition of a rational expression Q cannot equal 0. The denominator of a rational expression cannot equal 0 because division by 0 is undefined. Since we are solving to find values that make the expression undefined, we write the answer as “variable ≠ value”, not “variable = value or { }. For instance, in the rational expression the variable x can take on any real number value except 2. If x is 2, then the denominator becomes 2(2) − 4 = 0, making the expression undefined. Thus, x cannot equal 2. We indicate this restriction by writing x ≠ 2. Denominator cannot equal 0 Slide 6.1-7

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Determining When a Rational Expression is Undefined Step 1: Set the denominator of the rational expression equal to 0. Step 2: Solve this equation. Step 3: The solutions of the equation are the values that make the rational expression undefined. The variable cannot equal these values. The numerator of a rational expression may be any real number. If the numerator equals 0 and the denominator does not equal 0, then the rational expression equals 0. Slide 6.1-8 Find the values of the variable for which a rational expression is undefined. (cont’d)

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Find any values of the variable for which each rational expression is undefined. Solution: never undefined Slide 6.1-9 Finding Values That Make Rational Expressions Undefined CLASSROOM EXAMPLE 2

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Objective 3 Write rational expressions in lowest terms. Slide 6.1-10

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A fraction such as is said to be in lowest terms. Write rational expressions in lowest terms. Fundamental Property of Rational Expressions If (Q ≠ 0) is a rational expression and if K represents any polynomial, where K ≠ 0, then Lowest Terms A rational expression (Q ≠ 0) is in lowest terms if the greatest common factor of its numerator and denominator is 1. This property is based on the identity property of multiplication, since Slide 6.1-11

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Solution: Write each rational expression in lowest terms. Slide 6.1-12 Writing in Lowest Terms CLASSROOM EXAMPLE 3

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Quotient of Opposites If the numerator and the denominator of a rational expression are opposites, as in then the rational expression is equal to −1. Writing a Rational Expression in Lowest Terms Step 1: Factor the numerator and denominator completely. Step 2: Use the fundamental property to divide out any common factors. Rational expressions cannot be written in lowest terms until after the numerator and denominator have been factored. Only common factors can be divided out, not common terms. Numerator cannot be factored. Slide 6.1-13 Write rational expressions in lowest terms. (cont’d)

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Solution: Write in lowest terms. Slide 6.1-14 Writing in Lowest Terms CLASSROOM EXAMPLE 4

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Write in lowest terms. Solution: Slide 6.1-15 Writing in Lowest Terms (Factors Are Opposites) CLASSROOM EXAMPLE 5

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Write each rational expression in lowest terms. or Slide 6.1-16 Writing in Lowest Terms (Factors Are Opposites) Solution: CLASSROOM EXAMPLE 6

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Objective 4 Recognize equivalent forms of rational expressions. Slide 6.1-17

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Recognize equivalent forms of rational expressions. When working with rational expressions, it is important to be able to recognize equivalent forms of an expressions. For example, the common fraction can also be written and Consider the rational expression The − sign representing the factor −1 is in front of the expression, even with fraction bar. The factor −1 may instead be placed in the numerator or in the denominator. Some other equivalent forms of this rational expression are and Slide 6.1-18

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is not an equivalent form of. The sign preceding 3 in the numerator of should be − rather than +. Be careful to apply the distributive property correctly. By the distributive property, can also be written Slide 6.1-19 Recognize equivalent forms of rational expressions. (cont’d)

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Write four equivalent forms of the rational expression. Solution: Slide 6.1-20 Writing Equivalent Forms of a Rational Expression CLASSROOM EXAMPLE 7

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