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6 .1 Rational Expressions

6 .1 Rational Expressions. Rational Expression – an expression in which a polynomial is divided by another nonzero polynomial. Examples of rational expressions x 2

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6 .1 Rational Expressions

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  1. 6.1 Rational Expressions Rational Expression – an expression in which a polynomial is divided by another nonzero polynomial. • Examples of rational expressions • x 2 • x 2x – 5 x – 5 • Domain = {x | x  0} Domain = {x | x  5/2} Domain = {x | x  5}

  2. Graph of a Rational Function • y = 1 • x • x y • -2 -1/2 • -1 -1 • -1/2 -2 • 0 Undefined • ½ 2 • 1 • ½ The graph does not cross the x = 0 line since x the graph is undefined there.. The line x = 0 is called a vertical asymptote. An Application: Modeling a train track curve.

  3. Multiplication and Division of Rational Expressions A • C = A 9x = 3 B • C B 3x2 x 5y – 10 = 5 (y – 2) = 5 = 1 10y - 20 10 (y – 2) 10 2 2z2 – 3z – 9 = (2z + 3) (z – 3) = 2z + 3 z2 + 2z – 15 (z + 5) (z – 3) z + 5 A2 – B2 = (A + B)(A – B) = (A – B) A + B (A + B)

  4. Negation/Multiplying by –1 -y – 2 4y + 8 y + 2 4y + 8 -y - 2 -4y - 8 - = OR

  5. Examples x3 – x x + 1 x – 1 x x2 – 25 x2 –10x + 25 x2 + 5x + 4 2x2 + 8x •  x2 – 25 2x2 + 8x x2 + 5x + 4 x2 –10x + 25 • (x3 – x) (x + 1) x(x – 1) = = (x + 5) (x – 5) • 2x(x + 4) (x + 4)(x + 1) • (x – 5) (x – 5) = = x (x2 – 1)(x + 1) x(x – 1) 2x (x + 5) (x + 1)(x – 5) = x (x + 1) (x – 1)(x + 1) x(x – 1) = (x + 1)(x + 1) = (x + 1)2 =

  6. (x + 1) (x –7) (x + 1) (x – 1) 1 x2 + x - 6 x – 2 3 • 1 (x + 3) (x – 2) x – 2 3 (x – 7) (x – 1) • (x + 3) 3 Check Your Understanding Simplify: x2 –6x –7 x2 -1 Simplify: 1 3 x - 2 x2 + x - 6 

  7. 6.2 Addition of Rational Expressions Adding rational expressions is like adding fractions With LIKE denominators: 1 + 2 = 3 8 8 8 x + 3x - 1 = 4x - 1 x + 2 x + 2 x + 2 x + 2 (2 + x) (2 + x) 3x2 + 4x - 4 3x2 + 4x -4 (3x2 + 4x – 4) (3x -2)(x + 2) = = = 1 (3x – 2)

  8. Adding with UN-Like Denominators • + 2 • x2 – 9 x + 3 • 1 + 2 • (x + 3)(x – 3) (x + 3) • 1 + 2 (x – 3) • (x + 3)(x – 3) (x + 3)(x – 3) • 1 + 2(x – 3) 1 + 2x – 6 2x - 5 • (x + 3) (x – 3) (x + 3) (x – 3) (x + 3) (x – 3) • + 1 • 8 • (3) (2) + 1 • 8 • + 1 • 8 • 7 • 8 = =

  9. x – 1 (x + 1)(x –1) = = 1 (x + 1) Subtraction of Rational Expressions To subtract rational expressions: Step 1: Get a Common Denominator Step 2: Combine Fractions DISTRIBUTING the ‘negative sign’ BE CAREFUL!! 2x - x + 1 x2 – 1 x2 - 1 2x – (x + 1) x2 -1 2x – x - 1 x2 -1 = =

  10. b b-1 2(b – 2) b-2 - b -b+1 2(b – 2) b-2 + b 2(b – 2) 2(-b+1) 2(b – 2) + -1 2 -1(b – 2) 2(b – 2) b –2b+2 2(b – 2) -b + 2 2(b – 2) = = = Check Your Understanding Simplify: b b-1 2b - 4 b-2 -

  11. x + 2 3x - 1 x x + 4 6.3 Complex Fractions A complex fraction is a rational expression that contains fractions in its numerator, denominator, or both. Examples: 1 5 4 7 x x2 – 16 1 x - 4 1 x 2 x2 + 3 x 1 x2 - 7/20

  12. 6.4 Division by a Monomial 3x2 + x 5x3 – 15x2 x 15x 4x2 + 8x – 12 5x2y + 10xy2 4x2 5xy 15A2 – 8A2 + 12 12A5 – 8A2 + 12 4A 4A

  13. Begin by writing the divisor and dividend in descending powers of x. Then, figure out how many times 3x divides into 6x3. Multiply. Divide: 6x3/3x = 2x2. 2x2 3x – 2 6x3 – x2 – 5x + 4 Multiply: 2x2(3x – 2) = 6x3 – 4x2. 6x3 – 4x2 Subtract 6x3 – 4x2 from 6x3 – x2 and bring down –5x. -1 3x2 – 5x Now, divide 3x2 by 3x to obtain x, multiply then subtract. Subtract -3x + 2 from -3x + 4, leaving a remainder of 2. 2 Multiply. Divide: 3x2/3x = x. 2x2 + x 3x – 2 6x3 – x2 – 5x + 4 -3x +2 Multiply: x(3x – 2) = 3x2 – 2x. 6x3 – 4x2 Subtract 3x2 – 2x from 3x2 – 5x and bring down 4. 3x2 – 5x Answer: 2x2 + x – 1 + 2 3x - 2 3x2 – 2x -3x + 4 Example: Divide 4 – 5x – x2 + 6x3 by 3x – 2. Polynomial Long Division

  14. More Long Division 3x -11 3x3 + 9x2 + 9x -11x2 - 5x - 3 -11x2 - 33x - 33 28x+30

  15. (2x – 1) (x - 2) (x + 1) 6.5-6.6 Rational Equations 3x = 3 x + 1 = 3 6 = x 2x – 1 x – 2 x - 2 x + 1 3x = 3(2x – 1) 3x = 6x – 3 -3x = -3 x = 1 x + 1 = 3 x = 2 6 = x (x + 1) 6 = x2 + x x2 + x – 6 = 0 (x + 3 ) (x - 2 ) = 0 x = -3 or x = 2 Careful! – What do You notice about the answer?

  16. (12x) 6 (x + 1) -3(x – 1) = 4x 6x + 6 –3x + 3 = 4x 3x + 9 = 4x -3x -3x 9 = x = Rational Equations Cont… To solve a rational equation: Step 1: Factor all polynomials Step 2: Find the common denominator Step 3: Multiply all terms by the common denominator Step 4: Solve x + 1 - x – 1 = 1 2x 4x 3

  17. (4x2) (x + 2)(x – 2) 4x + 4 = 3x2 3x2 - 4x - 4 = 0 (3x + 2) (x – 2) = 0 3x + 2 = 0 or x – 2 = 0 3x = -2 or x = 2 x = -2/3 or x = 2 3(x + 2) + 5(x – 2) = 12 3x + 6 + 5x – 10 = 12 8x – 4 = 12 + 4 + 4 8x = 16 x = 2 Other Rational Equation Examples 3 + 5 = 12 x – 2 x + 2 x2 - 4 1 + 1 = 3 x x2 4 3 + 5 = 12 x – 2 x + 2 (x + 2) (x – 2)

  18. Solve for p: • = 1 + 1 • F p q 1 x - 1 2(x – 3) x(x – 2) 3 x(x – 1)(x + 1) Check Your Understanding Simplify: x 1 x2 – 1 x2 – 1 1 3 x – 2 x 1 1 2 x(x – 1) x2 – 1 x(x + 1) Solve 6 1 x 2 3 2 2x – 1 x + 1 2 3 x x – 1 x + 2 x2 + x - 2 4 + - = 1 = 5 - + = + - -1/4 Try this one:

  19. 6.7 Proportions & Variation • Proportion equality of 2 ratios. Proportions are used to solve problems in everyday life. • If someone earns $100 per day, then how many dollars can the • person earn in 5 days? • 100 x (x)(1) = (100)(5) • 1 5 x = 500 • 2. If a car goes 210 miles on 10 gallons of gas, the car can go 420 miles on X gallons • 210 420 (210)(x) = (420)(10) • 10 x (210)(x) = 4200 • x = 4200 / 210 = 20 gallons • If a person walks a mile in 16 min., that person can walk a half mile in x min. • 16 x (x)(1) = ½(16) • 1 ½ x = 8 minutes = = =

  20. personheight treeheight personshadow treeshadow x 6ft 2.5 100 6 x = 2.5x = (100)(6) 2.5x = 600 2.5 2.5 x = 240 feet 100 ft 2 ½ ft The Shadow Problem Juan is 6 feet tall, but his shadow is only 2 ½ feet long. There is a tree across the street with a shadow of 100 feet. The sun hits the tree and Juan at the same angle to make the shadows. How tall is the tree?

  21. y = kx y is directly proportional to x. y varies directly with x k is the constant of proportionality Example: y = 9x (9 is the constant of proportionality) Let y = Your pay Let x – Number of Hours worked Your pay is directly proportional to the number of hours worked. 7.6 Direct Variation Example1: Salary (L) varies directly as the number of hours worked (H). Write an equation that expresses this relationship. Salary = k(Hours) L = kH Example 2: Aaron earns $200 after working 15 hours. Find the constant of proportionality using your equation in example1.. 200 = k(15) So, k = 200/15 = 13.33

  22. Inverse Variation y = k y is inversely proportional to x x y varies inversely as x Example: y varies inversely with x. If y = 5 when x = 4, find the constant of proportionality (k) 5 = k So, k = 20 4

  23. Direct Variation with Power y = kxn y is directly proportional to the nth power of x Example: Distance varies directly as the square of the time (t) Distance = kt2 D = kt2 Joint Variation y = kxp • y varies jointly as x and p

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