11.3 Simplifying Rational Expressions - Big Ideas Learning

11.3 Simplifying Rational ExpressionsHow can you simplify a rational expression?What are the excluded values of a rational expression?1ACTIVITY: Simplifying a Rational ExpressionWork with a partner.Sample: You can see that therational expressionsx 2 3xx—2x 3xand —are equivalent by graphing therelated functionsx 2 3xxandy —2x 3xy —.4yâ4x 2 à 3xx2yâŹ66COMMONCORERational ExpressionsIn this lesson, you will simplify rationalexpressions.Learning StandardA.SSE.2Chapter 11MSCA ALG1 PE 1103.indd 560Ź66Ź4Ź4Both functions have the same graph.Match each rational expression with its equivalent rational expression.Use a graphing calculator to check your answers.x2 xxb.—2x2x xc.1x 1B.x 1C. —a. —2A. —5606xà3x—2x 1x 1d. —2x 1xD.x 2 2x 1x 1x 1x 2x 1e. —1x 1E.—xx 1—Rational Equations and Functions12/17/13 2:08:35 PM

2ACTIVITY: Finding Excluded ValuesWork with a partner. Are the graphs ofx2 xxy —andy x 1exactly the same? Explain your plain theMeaningWhat does it meanfor a simplifiedexpression to havean excluded value?Ź66Ź4ACTIVITY: Simplifying and Finding Excluded ValuesWork with a partner. Simplify each rational expression, if possible. Thencompare the excluded value(s) of the original expression with the excludedvalue(s) of the simplified expression.x 2 2xxb. —2x 2 4x 4x 2e. —2a. —2d. —x2x 2xc. —x2xx 2x 4f. —21x 14. IN YOUR OWN WORDS How can you simplify a rational expression? Whatare the excluded values of a rational expression? Include the followingrational expressions in your answer.x(x 1)xa. —x 2 3x 2x 2b. —x 3x 9c. —2Use what you learned about simplifying rational expressions tocomplete Exercises 3–5 on page 564.Section 11.3Simplifying Rational Expressions561

11.3 LessonLesson TutorialsA rational expression is an expression that can be written as a fractionwhose numerator and denominator are polynomials. Values that makethe denominator of the expression zero are excluded values.Key Vocabularyrational expression,p. 562simplest form of arational expression,p. 562Simplifying Rational ExpressionsA rational expression is in simplest form when the numeratorand denominator have no common factors except 1. Tosimplify a rational expression, factor the numerator anddenominator and divide out any common factors.WordsStudy TipYou can see why youcan divide out commonfactors by rewriting theexpression.acbcab— —Let a, b, and c be polynomials,where b, c 0.Algebra —c —b 1 —bcaacbca a cb cab— — —EXAMPLE1Example2(x 1) 2— —; x 15(x 1) 5Simplifying Rational ExpressionsSimplify each rational expression, if possible. State the excluded value(s). 122x2 2 32 x xDivide out the common factor.6xSimplify.a. —2 — —2The excluded value is x 0.nn 8b. —The expression is in simplest form. The excluded value is n 8.Study Tip3y 26y(y 7) 3 y y2 3 y (y 7)Divide out the common factors.y2(y 7)Simplify.c. — ——Make sure you findexcluded values usingthe original expression. —The excluded values are y 0 and y 7.Simplify the rational expression, if possible. State the excluded value(s).Exercises 3 – 8562Chapter 115y 32y1. —2Rational Equations and Functions2.8x(x 1)12x—23.m 1m(m 3)—

2EXAMPLESimplifying Rational ExpressionsSimplify each rational expression, if possible. State the excludedvalue(s).1 z2z 1(1 z)(1 z)z 1Difference of Two Squares Pattern (z 1)(1 z)z 1Rewrite 1 z as (z 1). —— (z 1)(1 z)z 1Divide out the common factor. z 1Simplify.a. — —— ——The excluded value is z 1.c 2 c 12c c 20(c 4)(c 3)(c 4)(c 5)Factor. Divide out the common factor.c 3c 5Simplify.b. — ——2 —The excluded values are c 4 and c 5.3EXAMPLEReal-Life ApplicationIn general, as the surface area to volume ratio of a substance increases,it reacts faster with other substances. Write and simplify this ratio for ablock of ice that has the shape shown.2xSurface areaVolume2(x 2) 4(2x 2)x(x)(2x)—— ——Write an expression.510x 22xSimplify. Divide out the common factors.5xSimplify. —3 —xxSimplify the rational expression, if possible. State the excludedExercises 10–15value(s).2b 87b 284. —5.2a 64a 12a—26.z 2 6z 168 z——7. What is the surface area to volume ratio of a cube-shapedsubstance with edge length x ?Section 11.3Simplifying Rational Expressions563

11.3 ExercisesHelp with Homework— 1x 3 x1. VOCABULARY Is — a rational expression? Explain.2. REASONING Why is it necessary to state excluded values of a rational expression?6) 39 (- 3) 3 (- 9) 4 (- 1)9 (-Simplify the rational expression, if possible. State the excluded value(s).115y 35y5. —4t 22t(t 11)8. —3618x4. —29w 312w7. —3. —6. —49. ERROR ANALYSIS Describe and correctthe error in stating the excluded value(s).n 1n 116x 2y24xy x3x (x 3)xx 3 ——2The excluded value is x 3.Simplify the rational expression. State the excluded value(s).3b 98b 2411. —4 y2y 3y 1014. ——22 10. —13. ——26a 2 12a9a 18a5 2z2z 512. —32n2 5n 6n 8n 1515. ——323x 3 12x6x 24x 24x(x 2)(x 5)(x 2)(5 x)16. WRITING Is —— in simplest form?Explain.17. RECYCLING You hang recycling posters onbulletin boards at your school. Simplify thedimensions of the poster.564Chapter 11Rational Equations and Functions(x à 3) 3(x à 3) 2x 2 Ź 3x2x Ź 6

Write and simplify a rational expression for the ratio of the perimeter of the figureto its area.19.18.20.2x4xxá3xá3xá12x2x á 24x21. OPEN-ENDED Write a rational expression whose excluded values are 3 and 5.x2 4x 222. WRITING Is — equivalent to x 2? Justify your answer.x in.(x à 4) in.23. PROBLEM SOLVING The candles shown have thesame volume. Write and simplify an expressionh in.for the height of the cone-shaped candle.Sandbox B2x in.Sandbox A24. SANDBOX The area of Sandbox Bis 4 square feet greater than thearea of Sandbox A. Write andsimplify an expression for thewidth w of Sandbox B.w ft4x ft(2x à 3) ft(2x à 1) ft4x 12x 1Find two polynomials whose simplified ratio is —25.and whose sum is 6x 2 12x. Explain your reasoning.Graph the function. Is the domain discrete or continuous? (Section 5.2)26.27.InputMonths, xOutputHeight of Plant, y (inches)211.32422.13632.9InputBoxes, xOutputNumber of Shoes, y128. MULTIPLE CHOICE Consider f (x) 2x 4. What is the value of x sothat f (x) 8? (Section 5.4)A 2 B 4 C 6 Section 11.3D 7 Simplifying Rational Expressions565

EXAMPLE 1 Simplifying Rational Expressions Exercises 3 – 8 Study Tip You can see why you can divide out common factors by rewriting the expression. ac — bc a — b c — c a — b 1 a — b Study Tip Make sure you fi nd excluded values using the original expression. Simplifying Rational E