Section 9.4 Rational Expressions

Rational Expressions Factor. ALGEBRA 2 LESSON 9-4 (For help, go to Lesson 5-4 and Skills Handbook page 843.) Factor. 1. 2x2 – 3x + 1    2. 4x2 – 9    3. 5x2 + 6x + 1    4. 10x2 – 10 Multiply or divide. 5. •      6.  •     7.  •     8.  • 9. ÷ 4     10.  ÷     11.  ÷      12.  ÷ 3 8 5 6 1 2 4 16 7 9 15 Check Skills You’ll Need 9-4

Check: (2x – 1)(x – 1) = 2x2 – 2x – 1x + 1 = 2x2 – 3x + 1 Rational Expressions ALGEBRA 2 LESSON 9-4 Solutions 1. 2x2 – 3x + 1 = (2x – 1)(x – 1) Check: (2x – 1)(x – 1) = 2x2 – 2x – 1x + 1 = 2x2 – 3x + 1 3. 5x2 + 6x + 1 = (5x + 1)(x + 1) Check: (5x + 1)(x + 1) = 5x2 + 5x + 1x + 1 = 5x2 + 6x + 1 2. 4x2 – 9 = (2x)2 – 32 = (2x + 3)(2x – 3) 4. 10x2 – 10 = 10(x2 – 1) = 10(x2 – 12) = 10(x + 1)(x – 1) 6. • = = = 8. • = = 10. ÷ = • = = = or 1 12. ÷ = • = = = 1 2 4 6 1 • 4 2 • 6 1 • 2 • 2 2 • 2 • 3 / 3 5 7 2 • 3 5 • 7 35 15 8 5 • 8 4 • 15 5 • 2 • 4 4 • 5 • 3 3 • 2 4 • 1 2 • 2 • 1 5. • = = = 7. • = = = 9. ÷ 4 = • = = 11. ÷ = • = = = 3 8 5 6 3 • 5 8 • 6 8 • 3 • 2 16 / 2 8 • 2 3 • 16 1 • 16 1 4 5 • 1 8 • 4 32 9 9 • 4 16 • 3 3 • 3 • 4 4 • 4 • 3 9-4

Simplify . State any restrictions on the variable. Rational Expressions ALGEBRA 2 LESSON 9-4 Simplify . State any restrictions on the variable. x2 – 6x – 16 x2 + 5x + 6 = Factor the polynomials. Notice x = –3 or –2. / x2 – 6x – 16 x2 + 5x + 6 (x – 8)(x + 2) (x + 3)(x + 2) = Divide out common factors. (x – 8)(x + 2) (x + 3)(x + 2) 1 = x – 8 x + 3 The simplified expression for x = –3 or –2. x + 8 x + 3 / The restrictions on x are needed to prevent the denominator of the original expression from being zero. Quick Check 9-4

Use the formulas for volume and surface area of a sphere. Rational Expressions ALGEBRA 2 LESSON 9-4 Quick Check 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. Use the formulas for volume and surface area of a sphere. Volume (V) = • r 3 Surface Area (S.A.) = 4 r 2 4 3 Sphere with radius r Sphere with radius 2r = = Write a ratio. V S.A. 4 3 r 3 4 r 2 4 3 (r ) 3 4 (r )2 =      = Substitute for r. (2r ) 3 4 (2r )2 = = Simplify. r 3 2r 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. 9-4

Multiply and . State any restrictions on the variable. Rational Expressions ALGEBRA 2 LESSON 9-4 Multiply and . State any restrictions on the variable. 3x2 + 5x – 2 x – 5 x2 – 25 3x2 – 7x + 2 • = • Factor. 3x2 + 5x – 2 x – 5 x2 – 25 3x2 – 7x + 2 (3x – 1)(x + 2) (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 = (x + 2)(x + 5) x – 2 = x2 + 7x + 10 x – 2 The product is for x 5, 2, or . x2 + 7x + 10 x – 2 1 3 = / Quick Check 9-4

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

Simplify. State any restrictions on the variable. 1. 2. Rational Expressions ALGEBRA 2 LESSON 9-4 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. ÷ x2 + x – 6 x2 + 3x ; x = –3, 0 / x – 2 x 4x2– 25 2x2 + 3x – 20 ; x = –4, / 2x + 5 x + 4 5 2 x2 + 6x – 7 x2 + 5x 3x2 + 16x + 5 2x2 + 7x – 9 ; x = –5, – , 0, 1 / 3x2 + 22x + 7 2x2 + 9x 9 2 y 2 + 5y + 4 y 2 – 49 2y 2 + 5y – 12 y 2 + 9y + 14 ; y = –7, –4, –2, , 7 / 3 2 y 2 + 3y + 2 2y 2 – 17y + 21 9-4