6 .1 rational expressions
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6.1 Rational Expressions
Rational Expression – an expression in which a polynomial is divided by another nonzero polynomial.
Examples of rational expressions
4 x 2x 2x – 5 x – 5
Domain = {x | x 0} Domain = {x | x 5/2} Domain = {x | x 5}
Graph of a Rational Function
y = 1 x
x y-2 -1/2-1 -1-1/2 -20 Undefined½ 21 12 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.
Multiplication and Division of Rational Expressions
A • C = A 9x = 3B • C B 3x2 x
5y – 10 = 5 (y – 2) = 5 = 110y - 20 10 (y – 2) 10 2
2z2 – 3z – 9 = (2z + 3) (z – 3) = 2z + 3z2 + 2z – 15 (z + 5) (z – 3) z + 5
A2 – B2 = (A + B)(A – B) = (A – B)A + B (A + B)
Examples
x3 – x x + 1x – 1 x
•
(x3 – x) (x + 1) x(x – 1)=
x (x2 – 1)(x + 1) x(x – 1)
=
= x (x + 1) (x – 1)(x + 1) x(x – 1)
= (x + 1)(x + 1) = (x + 1)2
x2 – 25 x2 –10x + 25x2 + 5x + 4 2x2 + 8x
=x2 – 25 2x2 + 8xx2 + 5x + 4 x2 –10x + 25
•
=(x + 5) (x – 5) • 2x(x + 4)(x + 4)(x + 1) • (x – 5) (x – 5)
=2x (x + 5)(x + 1)(x – 5)
Check Your Understanding
Simplify:
x2 –6x –7 x2 -1
Simplify:
1 3x - 2 x2 + x - 6
(x + 1) (x –7)(x + 1) (x – 1)
(x – 7)(x – 1)
1 x2 + x - 6x – 2 3•
1 (x + 3) (x – 2)x – 2 3
•
(x + 3) 3
6.2 Addition of Rational ExpressionsAdding 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)
Adding with UN-Like Denominators
3 + 14 8
(3) (2) + 18 8
6 + 18 8
7 8
1 + 2x2 – 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)
= =
Subtraction of Rational Expressions
2x - x + 1x2 – 1 x2 - 1
To subtract rational expressions:Step 1: Get a Common DenominatorStep 2: Combine Fractions DISTRIBUTING the ‘negative sign’
BE CAREFUL!!
=2x – (x + 1)x2 -1
= x – 1(x + 1)(x –1)
= 1(x + 1)
= 2x – x - 1x2 -1
Check Your Understanding
Simplify:
b b-12b - 4 b-2-
b b-12(b – 2) b-2
-
b -b+12(b – 2) b-2
+
b2(b – 2)
2(-b+1)2(b – 2)
+
b –2b+22(b – 2)
-b + 22(b – 2)= =
-1(b – 2)2(b – 2)
= -12
6.3 Complex Fractions
A complex fraction is a rational expression that contains fractions in its numerator, denominator, or both.
Examples:
15
47
xx2 – 16
1x - 4
1x
+ 2x2
3x
- 1x2
7/20 xx + 4
x + 23x - 1
6.4 Division by a Monomial3x2 + x 5x3 – 15x2
x 15x
4x2 + 8x – 12 5x2y + 10xy2
4x2 5xy
15A2 – 8A2 + 12 12A5 – 8A2 + 12 4A 4A
Polynomial Long DivisionExample: Divide 4 – 5x – x2 + 6x3 by 3x – 2.
Begin by writing the divisor and dividend in descending powers of x. Then, figure out how many times 3x divides into 6x3.
3x – 2 6x3 – x2 – 5x + 42x2 Divide: 6x3/3x = 2x2.
6x3 – 4x2
Multiply.
Multiply: 2x2(3x – 2) = 6x3 – 4x2.
Divide: 3x2/3x = x.
Now, divide 3x2 by 3x to obtain x, multiply then subtract.
3x – 2 6x3 – x2 – 5x + 46x3 – 4x2
2x2 + x
3x2 – 5x
Multiply.
Multiply: x(3x – 2) = 3x2 – 2x.
3x2 – 2xSubtract 3x2 – 2x from 3x2 – 5x and bring down 4.
+ 4-3x
– 5x Subtract 6x3 – 4x2 from 6x3 – x2 and bring down –5x.
3x2
Subtract -3x + 2 from -3x + 4, leaving a remainder of 2.
2
-3x +2
-1
Answer: 2x2 + x – 1 + 2 3x - 2
6.5-6.6 Rational Equations
3x = 3 x + 1 = 3 6 = x2x – 1 x – 2 x - 2 x + 1
(2x – 1)
3x = 3(2x – 1)3x = 6x – 3-3x = -3
x = 1
(x - 2)
x + 1 = 3
x = 2
(x + 1)
6 = x (x + 1)
6 = x2 + x
x2 + x – 6 = 0
(x + 3 ) (x - 2 ) = 0
x = -3 or x = 2Careful! – What doYou notice about theanswer?
Rational Equations Cont…To solve a rational equation:
Step 1: Factor all polynomialsStep 2: Find the common denominatorStep 3: Multiply all terms by the common denominatorStep 4: Solve
x + 1 - x – 1 = 1 2x 4x 3
(12x)
= 6 (x + 1) -3(x – 1) = 4x6x + 6 –3x + 3 = 4x
3x + 9 = 4x -3x -3x 9 = x
Other Rational Equation Examples
3 + 5 = 12x – 2 x + 2 x2 - 4
3 + 5 = 12x – 2 x + 2 (x + 2) (x – 2)
(x + 2)(x – 2)
3(x + 2) + 5(x – 2) = 12
3x + 6 + 5x – 10 = 12
8x – 4 = 12 + 4 + 4
8x = 16
x = 2
1 + 1 = 3x x2 4
(4x2)
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
Check Your Understanding
Simplify:x 1x2 – 1 x2 – 1
1 3x – 2 x
1 1 2x(x – 1) x2 – 1 x(x + 1)
Solve6 1x 2
3 22x – 1 x + 1
2 3 xx – 1 x + 2 x2 + x - 2
+
-
+ -
- = 1
=
+ =
1x - 1
2(x – 3)x(x – 2)
3x(x – 1)(x + 1)
4
5
-1/4
1 = 1 + 1F p q
Solve for p:Try this one:
6.7 Proportions & VariationProportion equality of 2 ratios. Proportions are used to solve problems in everyday life.
1. 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
3. 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
=
=
=
The Shadow ProblemJuan 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?
6ft
2 ½ ft
x
100 ft
6x
= 2.5100
2.5x = (100)(6)
2.5x = 6002.5 2.5
x = 240 feet
personheight treeheight
personshadow
treeshadow
7.6 Direct Variationy = kx y is directly proportional to x.
y varies directly with xk is the constant of proportionality
Example: y = 9x (9 is the constant of proportionality)Let y = Your payLet x – Number of Hours workedYour pay is directly proportional to the number of hours worked.
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
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
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