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Section 9.4 Rational Expressions

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(For help, go to Lesson 5-4 and Skills Handbook page 843.) ALGEBRA 2 LESSON 9-4 Rational Expressions Factor. 1.2x 2 – 3x + 1 2. 4x 2 – 9 3. 5x 2 + 6x + 1 4. 10x 2 – 10 Multiply or divide. 5. 6. 7. 8. 9. ÷ 4 10. ÷ 11. ÷ 12. ÷ 3838 5656 1212 4646 8383 2 16 2525 3737 5858 3434 1212 9 16 3434 5454 15 8 9-4 Check Skills You’ll Need

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ALGEBRA 2 LESSON 9-4 Rational Expressions 2.4x 2 – 9 = (2x)2 – 32 = (2x + 3)(2x – 3) 4.10x 2 – 10 = 10(x 2 – 1) = 10(x 2 – 12) = 10(x + 1)(x – 1) Solutions 1.2x 2 – 3x + 1 = (2x – 1)(x – 1) Check: (2x – 1)(x – 1) = 2x 2 – 2x – 1x + 1 = 2x 2 – 3x + 1 3.5x 2 + 6x + 1 = (5x + 1)(x + 1) Check: (5x + 1)(x + 1) = 5x 2 + 5x + 1x + 1 = 5x 2 + 6x + 1 5. = = = 7. = = = 9. ÷ 4 = = = 11. ÷ = = = = 3838 5656 3 5 8 6 3 5 8 3 2 5 16 / / 8383 2 16 8 2 3 16 1 16 3 16 1313 5858 5858 1414 5 1 8 4 5 32 9 16 3434 9 16 4343 9 4 16 3 3 3 4 4 4 3 / / / 3434 6. = = = 8. = = 10. ÷ = = = = or 1 12. ÷ = = = = 1212 4646 1 4 2 6 1 2 2 2 2 3 / / / / 1313 2525 3737 2 3 5 7 6 35 5454 15 8 5454 15 5 8 4 15 5 2 4 4 5 3 / / / / 2323 3434 1212 3434 2121 3 2 4 1 3 2 2 2 1 3232 / 1212 / 9-4

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Simplify. State any restrictions on the variable. ALGEBRA 2 LESSON 9-4 Rational Expressions x 2 – 6x – 16 x 2 + 5x + 6 The restrictions on x are needed to prevent the denominator of the original expression from being zero. = Factor the polynomials. Notice x = –3 or –2. / x 2 – 6x – 16 x 2 + 5x + 6 (x – 8)(x + 2) (x + 3)(x + 2) = Divide out common factors. (x – 8)(x + 2) (x + 3)(x + 2) 1 1 = x – 8 x + 3 The simplified expression for x = –3 or –2. x + 8 x + 3 / 9-4 Quick Check

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Compare the ratio of the volume to surface area of a sphere with radius r with the ratio of volume to surface area of a sphere with radius 2r. ALGEBRA 2 LESSON 9-4 Rational Expressions The ratio of volume to surface area is twice as great for a sphere whose radius is 2r than a sphere with a radius of r. Use the formulas for volume and surface area of a sphere. Volume (V) = r 3 Surface Area (S.A.) = 4 r 2 4343 = = Simplify. r3r3 2r 3 9-4 4343 (r ) 3(r ) 3 4 (r ) 2 = = Substitute for r. 4343 (2r ) 3(2r ) 3 4 (2r ) 2 Sphere with radius rSphere with radius 2r = = Write a ratio. V S.A. V S.A. 4343 r 3r 3 4 r 2 4343 r 3r 3 Quick Check

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ALGEBRA 2 LESSON 9-4 Rational Expressions Multiply and. State any restrictions on the variable. 3x 2 + 5x – 2 x – 5 x 2 – 25 3x 2 – 7x + 2 = Factor. 3x 2 + 5x – 2 x – 5 x 2 – 25 3x 2 – 7x + 2 (3x – 1)(x + 2) x – 5 (x + 5)(x – 5) (3x – 1)(x – 2) = Divide out common factors. (3x – 1)(x + 2) x – 5 (x + 5)(x – 5) (3x – 1)(x – 2) 1 11 1 = (x + 2)(x + 5) x – 2 = x 2 + 7x + 10 x – 2 9-4 The product is for x 5, 2, or. x 2 + 7x + 10 x – 2 1313 =/ Quick Check

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Divide by. State any restrictions on the variables. ALGEBRA 2 LESSON 9-4 Rational Expressions 3 – y (2x – 1)(x + 5) 6(y – 3) (2x – 1)(x – 7) ÷ = Multiply by the reciprocal. 3 – y (2x – 1)(x + 5) 6(y – 3) (2x – 1)(x – 7) 3 – y (2x – 1)(x + 5) (2x – 1)(x – 7) 6(y – 3) = Divide out common factors. 3 – y (2x – 1)(x + 5) (2x – 1)(x – 7) 6(y – 3) 1 1 1 –1 = (x + 5) (x – 7) 6 = 7 – x 6(x + 5) = Multiply. 7 – x 6x + 30 9-4 The quotient is for x = –5,, or 7, and y = 3. 7 – x 6x + 30 / / 1212 Quick Check

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ALGEBRA 2 LESSON 9-4 Rational Expressions Simplify. State any restrictions on the variable. 1.2. 3.Multiply. State any restrictions on the variable. 4.Divide. State any restrictions on the variable. ÷ x 2 + x – 6 x 2 + 3x 4x 2 – 25 2x 2 + 3x – 20 x 2 + 6x – 7 x 2 + 5x 3x 2 + 16x + 5 2x 2 + 7x – 9 y 2 + 5y + 4 y 2 – 49 2y 2 + 5y – 12 y 2 + 9y + 14 ; x = –3, 0/ x – 2 x ; x = –5, –, 0, 1 / 3x 2 + 22x + 7 2x 2 + 9x 9292 ; y = –7, –4, –2,, 7 / 3232 y 2 + 3y + 2 2y 2 – 17y + 21 9-4 ; x = –4, / 2x + 5 x + 4 5252

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