Section 2.7 Notes Name: Date: Polynomial and Rational Inequalities

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1 Section.7 Notes Name: Date: Precalculus Polynomial and Rational Inequalities At the beginning of this unit we solved quadratic inequalities by using an analysis of the graph of the parabola combined with test intervals. Our basic technique was to: 1.) Find the zeros (through factoring or quadratic formula)..) Make a sketch using the zeros and end behavior (whether the parabola opened up or down). 3.) Shade the proper region(s) using the inequality symbol & test values in each region to confirm our interval answer. We can also solve polynomial and rational inequalities in a similar way. Solving Polynomial Inequalities Graphically using Multiplicity Given f ( ) is a polynomial in standard form: 1. Write f in completely factored form.. Plot real zeros on the -ais, noting their multiplicity. If the multiplicity is odd the function will go through the ais. If the multiplicity is even the function will bounce off of the ais. 3. Use the end-behavior to determine the sign of f in outermost intervals. 4. State the solution in interval notation. Special tip: Use the y-intercept as a quick check of the sign of its interval. Eample 1: < + Eample :

2 An alternate way to solve polynomial inequalities graphically using multiplicity Given f ( ) is a polynomial in standard form: 1. Write f in completely factored form.. Plot real zeros on the -ais, noting their multiplicity. 3. Test a value in each interval to determine if f ( ) < 0 or f ( ) > 0 in that interval. 4. State the final solution in interval notation. Special tip: Use the y-intercept as a quick check of the sign of its interval. Eample 3:

3 Section.7 Notes Name: Date: Precalculus Polynomial and Rational Inequalities Vertical Asymptotes and Multiplicities The cross and bounce concept used for polynomial graphs can also be applied to rational graphs, particularly when viewed in terms of sign changes in the dependent variable. Odd Multiplicity ( ) f 1 = + Vertical Asymptote(s): = Even Multiplicity 1 ( 1) ( ) = g Vertical Asymptote(s): = 1 Combination 1 ( + )( 1) ( ) = k Vertical Asymptote(s): = & = 1 If zero of denominator has multiplicity then graph If zero of denominator has multiplicity then graph Find the vertical asymptote(s) and use their multiplicity to state whether the function will change sign from one side of the asymptote(s) to the other. Confirm your answer afterwards using your graphing calculator. Eample 1: f ( ) = Vertical asymptote(s) Multiplicity Change/No Change Eample : f ( ) + = Vertical asymptote(s) Multiplicity Change/No Change Solving Rational Inequalities Graphically using Multiplicity

4 Given f ( ) is a rational function in standard form: 1. Write f in completely factored form.. Identify the zeros & the vertical asymptotes. These are the locations that break the -ais into intervals. Note their multiplicity. If the multiplicity is odd the function will change sign. If the multiplicity is even there will be no change in sign. 3. a. Use the y-intercept rather than the end-behavior to determine the sign of f and then work outward using the change/no change approach through analyzing the multiplicity of neighboring zeros. OR b. Test a value in each interval to determine if f ( ) < 0 or f ( ) > 0 in that interval. 4. State the solution in interval notation. Special tip: Remember when plotting the zeros of the denominator (a.k.a. the locations for the vertical asymptotes) to ALWAYS use open circles since these are not values in the domain of the function. Eample Eample 5: 3 0 9

5 Section.7 HW Name: Date: Precalculus Polynomial and Rational Inequalities #1 6: Solve each polynomial inequality using either a graphical analysis or a test interval method. Write all answers in interval notation. 1. ( + 3)( 5) < 0. ( + 1) ( 4) < > >

6 #7 10: Solve each rational inequality using either a graphical analysis or a test interval method. Write all answers in interval notation <

( ) = 1 x. g( x) = x3 +2

( ) = 1 x. g( x) = x3 +2 Rational Functions are ratios (quotients) of polynomials, written in the form f x N ( x ) and D x ( ) are polynomials, and D x ( ) does not equal zero. The parent function for rational functions is f x