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Chapter 8:Rational Functions
Big ideas: Performing operations with rational expressions Solving rational equations
VOCABULARY
Inverse variation ~ The relationship of two variables x and y if there is a nonzero number a such that y =
Constant of variation ~ The nonzero constant a in a direct variation equation y=ax, an inverse variation equation y=, or a joint variation equation z=axy
Joint variation ~ A relationship that occurs when a quantity varies directly with the product of two or more other quantities.
EXAMPLE 1 Classify direct and inverse variation
Tell whether x and y show direct variation, inverse variation, or neither.
Given Equation Rewritten Equation Type of Variation
a. xy = 7 y =7
xInverse
b. y = x + 3 Neither
y
4c. = x y = 4x Direct
EXAMPLE 2 Write an inverse variation equation
The variables x and y vary inversely, and y = 7 when x = 4. Write an equation that relates x and y. Then find y when x = –2 .
y =a
xWrite general equation for inverse variation.
Substitute 7 for y and 4 for x. 7 =a
4
28 = a Solve for a.
The inverse variation equation is y =28x
When x = –2, y = 28–2
= –14.
ANSWER
Essential question
What are the differences between direct, inverse and joint variation?
~ y varies directly with x if y=ax for nonzero constant a
~ y varies inversely with x if xy = a for a nonzero constant a
~ z varies jointly with x and y if z = axy for nonzero constant a
VOCABULARY
Simplified form of a rational expression ~ A rational expression in which the numerator and denominator have no common factors other than 1 and -1
Reciprocal ~ The multiplicative inverse of any nonzero number
EXAMPLE 1 Simplify a rational expression
x2 – 2x – 15x2 – 9
Simplify :
x2 – 2x – 15x2 – 9
(x +3)(x –5)(x +3)(x –3)= Factor numerator and denominator.
(x +3)(x –5)(x +3)(x –3)
= Divide out common factor.
Simplified form
SOLUTION
x – 5x – 3=
ANSWERx – 5x – 3
EXAMPLE 3 Standardized Test Practice
SOLUTION
8x 3y2x y2
7x4y3
4y56x7y4
8xy3= Multiply numerators
and denominators.
8 7 x x6 y3 y
8 x y3 = Factor and divide out
common factors.
7x6y= Simplified form
The correct answer is B.ANSWER
EXAMPLE 4 Multiply rational expressions
Multiply: x2 + x – 203x
3x –3x2
x2 + 4x – 5
x2 + x – 203x
3x –3x2
x2 + 4x – 5
3x(1– x)(x –1)(x +5)
=(x + 5)(x – 4)
3xFactor numerators and denominators.
3x(1– x)(x + 5)(x – 4)=
(x –1)(x + 5)(3x)Multiply numerators and denominators.
3x(–1)(x – 1)(x + 5)(x – 4)=
(x – 1)(x + 5)(3x)Rewrite 1– x as (– 1)(x – 1).
3x(–1)(x – 1)(x + 5)(x – 4)=
(x – 1)(x + 5)(3x)Divide out common factors.
SOLUTION
EXAMPLE 5 Multiply a rational expression by a polynomial
Multiply: x + 2x3 – 27
(x2 + 3x + 9)
x + 2x3 – 27
(x2 + 3x + 9)
Write polynomial as a rational expression.
=x + 2x3 – 27
x2 + 3x + 91
(x + 2)(x2 + 3x + 9)
(x – 3)(x2 + 3x + 9)= Factor denominator.
(x + 2)(x2 + 3x + 9)
(x – 3)(x2 + 3x + 9)= Divide out common factors.
=x + 2x – 3
Simplified form
SOLUTION
ANSWERx + 2x – 3
GUIDED PRACTICE for Examples 3, 4 and 5
Multiply the expressions. Simplify the result.
3x5 y2
8xy
6xy2
9x3y8.
3x5 y2
2xy
6xy2
9x3y
18x6y4
72x4y2= Multiply numerators
and denominators.
Factor and divide out common factors.
Simplified form
18 x4 y2 x2 y2
18 4 x4 y2 =
=x2y2
4
SOLUTION
GUIDED PRACTICE for Examples 3, 4 and 5
9.
x + 3
2x2
2x2 – 10xx2– 25
Factor numerators and denominators.
2x(x –5) (x + 3)=
(x –5)(x + 5)2x (x)Multiply numerators and denominators.
x + 3=
x(x + 5)
Divide out common factors.
x + 3
2x2
2x2 – 10xx2– 25
2x(x –5)(x –5)(x +5)
=x + 3
2x (x)
= 2x(x –5) (x + 3)(x –5)(x + 5)2x (x)
Simplified form
SOLUTION
GUIDED PRACTICE for Examples 3, 4 and 5
10.
Factor denominators.
Multiply numerators and denominators.
x + 5=x – 1
Divide out common factors.
x2 +x + 1x + 5x3– 1
Simplified form
x2 +x + 1x + 5x3– 1
=x2 +x + 1x + 5
(x – 1) (x2 +x + 1) 1
=(x + 5) (x2 +x + 1)
(x – 1) (x2 +x + 1)
(x + 5) (x2 +x + 1)
(x – 1) (x2 +x + 1)=
SOLUTION
EXAMPLE 6 Divide rational expressions
Divide : 7x2x – 10
x2 – 6xx2 – 11x + 30
7x2x – 10
x2 – 6xx2 – 11x + 30
7x2x – 10 x2 – 6x
x2 – 11x + 30= Multiply by reciprocal.
7x2(x – 5)
=(x – 5)(x – 6)
x(x – 6)
=7x(x – 5)(x – 6)
2(x – 5)(x)(x – 6) Divide out common factors.
Factor.
7
2= Simplified form
SOLUTION
ANSWER 7
2
EXAMPLE 7 Divide a rational expression by a polynomial
Divide : 6x2 + x – 154x2
(3x2 + 5x)
6x2 + x – 154x2
(3x2 + 5x)
6x2 + x – 154x2 3x2 + 5x
= 1 Multiply by reciprocal.
(3x + 5)(2x – 3)
4x2=
x(3x + 5)1 Factor.
Divide out common factors.
Simplified form2x – 34x3
=
(3x + 5)(2x – 3)=
4x2(x)(3x + 5)
SOLUTION
ANSWER2x – 34x3
GUIDED PRACTICE for Examples 6 and 7
Divide the expressions. Simplify the result.
4x5x – 20
x2 – 2xx2 – 6x + 8
11.
4x5x – 20
x2 – 2xx2 – 6x + 8
Multiply by reciprocal.
Divide out common factors.
Factor.
Simplified form
4x5x – 20 x2 – 2x
x2 – 6x + 8=
4(x)(x – 4)(x – 2)
5(x – 4)(x)(x – 2)=
4(x)(x – 4)(x – 2)
5(x – 4)(x)(x – 2)=
45
=
SOLUTION
GUIDED PRACTICE for Examples 6 and 7
2x2 + 3x – 5
6x(2x2 + 5x)12.
2x2 + 3x – 5
6x(2x2 + 5x)
Multiply by reciprocal.
Divide out common factors.
Factor.
Simplified form
=2x2 + 3x – 5
6x (2x2 + 5x)1
(2x + 5)(x – 1)
6x(x)(2 x + 5)=
(2x + 5)(x – 1)
6x(x)(2 x + 5)=
x – 1
6x2=
SOLUTION
Essential question
What are the steps for multiplying & dividing rational expressions?
Multiply: multiply the numerators / multiply the denominators then
simplifyDivide: multiply the first
expression by the reciprocal of the second expression, then follow the
rules for multiplication
Essential question
What are the steps for adding or subtracting
rational expressions with different denominators?
EXAMPLE 1 Add or subtract with like denominators
Perform the indicated operation.
74x
+3
4xa. 2x
x + 6– 5
x + 6b.
SOLUTION
74x
+3
4xa. =
7 + 34x
104x
=52x= Add numerators and
simplify result.
x + 6 2x – 5=2x
x + 65
x + 6–b. Subtract numerators.
GUIDED PRACTICE for Example 1
Perform the indicated operation and simplify.
16x
a. 712x
+5
12x=
7 – 512x
= 212x
= Subtract numerators and simplify results .
1 x2
b. 2 3x2
+1
3x2=
2 + 13x2 =
33x2
= Add numerators and simplify results.
c. 4x x–2
–x
x–2=
4x–xx–2
= 3xx–2
= Subtract numerators.
d. 4xx2+1
+ 2
x2+12x2+2x2+1
=2(x2+1)
x2+1= = 2
Factor numerators and simplify results .
3x3x – 2
EXAMPLE 2 Find a least common multiple (LCM)
Find the least common multiple of 4x2 –16 and 6x2 –24x + 24.
SOLUTION
STEP 1
Factor each polynomial. Write numerical factors asproducts of primes.
4x2 – 16 = 4(x2 – 4) = (22)(x + 2)(x – 2)
6x2 – 24x + 24 = 6(x2 – 4x + 4) = (2)(3)(x – 2)2
EXAMPLE 2 Find a least common multiple (LCM)
STEP 2
Form the LCM by writing each factor to the highest power it occurs in either polynomial.
LCM = (22)(3)(x + 2)(x – 2)2 = 12(x + 2)(x – 2)2
EXAMPLE 3 Add with unlike denominators
Add: 9x2
7+
x3x2 + 3x
SOLUTION
To find the LCD, factor each denominator and write each factor to the highest power it occurs. Note that 9x2 = 32x2 and 3x2 + 3x = 3x(x + 1), so the LCD is 32x2 (x + 1) = 9x2(x 1 1).
Factor second denominator.7
9x2
x3x2 + 3x
=7
9x2 +3x(x + 1)
x+
79x2
x + 1x + 1
+3x(x + 1)
x3x3x
LCD is 9x2(x + 1).
EXAMPLE 3 Add with unlike denominators
Multiply.
Add numerators.3x2 + 7x + 79x2(x + 1)
=
7x + 79x2(x + 1)
3x29x2(x + 1)
+=
EXAMPLE 4 Subtract with unlike denominators
Subtract: x + 22x – 2
–2x –1x2 – 4x + 3
–
SOLUTION
x + 22x – 2
–2x –1x2 – 4x + 3
–
x + 22(x – 1)
– 2x – 1(x – 1)(x – 3)
–= Factor denominators.
x + 22(x – 1)
=x – 3x – 3
– – 2x – 1(x – 1)(x – 3) 2
2 LCD is 2(x 1)(x 3).
x2 – x – 62(x – 1)(x – 3)
– 4x – 22(x – 1)(x – 3)
–= Multiply.
EXAMPLE 4 Subtract with unlike denominators
x2 – x – 6 – (– 4x – 2)2(x – 1)(x – 3)
= Subtract numerators.
x2 + 3x – 42(x – 1)(x – 3)
= Simplify numerator.
Factor numerator. Divide out common factor.
Simplify.
=(x –1)(x + 4)
2(x – 1)(x – 3)
x + 42(x –3)
=
GUIDED PRACTICE for Examples 2, 3 and 4
Find the least common multiple of the polynomials.
5. 5x3 and 10x2–15x
STEP 1
Factor each polynomial. Write numerical factors asproducts of primes.
5x3 = 5(x) (x2)
10x2 – 15x = 5(x) (2x – 3)
STEP 2
Form the LCM by writing each factor to the highest power it occurs in either polynomial.
LCM = 5x3 (2x – 3)
GUIDED PRACTICE for Examples 2, 3 and 4
Find the least common multiple of the polynomials.
6. 8x – 16 and 12x2 + 12x – 72
STEP 1
Factor each polynomial. Write numerical factors asproducts of primes.
8x – 16 = 8(x – 2) = 23(x – 2)
12x2 + 12x – 72 = 12(x2 + x – 6) = 4 3(x – 2 )(x + 3)
STEP 2
Form the LCM by writing each factor to the highest power it occurs in either polynomial.
= 24(x – 2)(x + 3)
LCM = 8 3(x – 2)(x + 3)
GUIDED PRACTICE for Examples 2, 3 and 4
4x3
–717.
SOLUTION
4x3
–71
4x3
77
71–
4x4x
LCD is 28x
4x7(4x)
214x(7)
–Multiply
Simplify21 – 4x
28x=
GUIDED PRACTICE for Examples 2, 3 and 4
13x2
+x
9x2 – 12x8.
SOLUTION
13x2
+x
9x2 – 12x
=1
3x2+
x3x(3x – 4)
=1
3x2
3x – 4 3x – 4
+ x3x(3x – 4)
x x
3x – 4 3x2 (3x – 4)
+x2
3x2 (3x – 4 )=
Factor denominators
LCD is 3x2 (3x – 4)
Multiply
GUIDED PRACTICE for Examples 2, 3 and 4
3x – 4 + x2 3x2 (3x – 4)
x2 + 3x – 4 3x2 (3x – 4)
Add numerators
Simplify
GUIDED PRACTICE for Examples 2, 3 and 4
xx2 – x – 12
+ 512x – 48
9.
SOLUTION
xx2 – x – 12
+ 512x – 48
=x
(x+3)(x – 4)+
5
12 (x – 4)
x(x + 3)(x – 4)
=1212
+5
12(x – 4)
x + 3
x + 3
5(x + 3)
12(x + 3)(x – 4)12x
12(x + 3)(x – 4)= +
Factor denominators
LCD is 12(x – 4) (x+3)
Multiply
GUIDED PRACTICE for Examples 2, 3 and 4
12x + 5x + 1512(x + 3)(x – 4)
=
17x + 1512(x +3)(x + 4)
=
Add numerators
Simplify
GUIDED PRACTICE for Examples 2, 3 and 4
x + 1x2 + 4x + 4
– 6x2 – 4
10.
SOLUTION
x + 1x2 + 4x + 4
– 6x2 – 4
=x + 1
(x + 2)(x + 2)
6
(x – 2)(x + 2) –
x + 1
(x + 2)(x + 2)=
x – 2x – 2
–6
(x – 2)(x + 2)x + 2x + 2
=x2 – 2x + x – 2
(x + 2)(x + 2)(x – 2) –
6x + 12
(x – 2)(x + 2)(x + 2)
Factor denominators
LCD is (x – 2) (x+2)2
Multiply
GUIDED PRACTICE for Example 2, 3 and 4
=x2 – 2x + x – 2 – (6x + 12)
(x + 2)2(x – 4)
=x2 – 7x – 14
(x + 2)2 (x – 2)
Subtract numerators
Simplify
EXAMPLE 6 Simplify a complex fraction (Method 2)
Simplify:
5x + 4
1x + 4
+ 2x
SOLUTION
The LCD of all the fractions in the numerator and denominator is x(x + 4).
5x + 4
1x + 4
+ 2x
5x + 4
1x + 4
+ 2x
=x(x+4)
x(x+4)Multiply numerator and denominator by the LCD.
x + 2(x + 4)5x
= Simplify.
Simplify.5x
3x + 8=
GUIDED PRACTICE for Examples 5 and 6
x
6
x
3–
x
57
10–
– 5x 3 (2x – 7)
=
Multiply numberator and denominator by the LCD
Simplify
x
6
x
3–
x
57
10–
11.
x
6
x
3–
x
57
10–
3030
=
GUIDED PRACTICE for Examples 5 and 6
2
x–
2
x+
4
3
12.
2 – 4x2 + 3x
=
2 (1 – 2x )
2 + 3x= Simplify
Simplify
2
x–
2
x+
4
3=
2
x–
2
x+
4
3
xx Multiply numberator and
denominator by the LCD
GUIDED PRACTICE for Examples 5 and 6
3x + 5
2x – 3
+ 1x + 5
13.
3x – 33x + 7
=
3(x – 3)
3x + 7=
Multiply numberator and denominator by the LCD
Simplify
Simplify
3x + 5
2x – 3
+ 1x + 5
=
3x + 5
2x – 3
+ 1x + 5
(x + 5)(x – 3)
(x + 5)(x – 3)
Essential question
What are the steps for adding or subtracting rational expressions with
different denominators?
1. Find the least common denominator2. Rewrite each fraction using the
common denominator3. Add or subtract
4. Simplify
VOCABULARY
Cross mulitplying ~ A method for solving a simple rational equation for which each side of the equation is a single rational expression
Extraneous solution
EXAMPLE 1 Solve a rational equation by cross multiplying
Solve: 3x + 1
= 94x + 1
3x + 1
= 94x + 1
Write original equation.
3(4x + 5) = 9(x + 1) Cross multiply.
12x + 15 = 9x + 9 Distributive property
3x + 15 = 9 Subtract 9x from each side.
3x = – 6 Subtract 15 from each side.
x = – 2 Divide each side by 3.
The solution is –2. Check this in the original equation.
ANSWER
EXAMPLE 3 Standardized Test Practice
SOLUTION
5x
+74
= – 9x
– 9x
4x4x 5x
+74( )=
Write original equation.
Multiply each side by the LCD, 4x.
20 + 7x = –36 Simplify.
7x = – 56 Subtract 20 from each side.
x = – 8 Divide each side by 7.
EXAMPLE 4 Solve a rational equation with two solutions
Solve: 1 – 8x – 5
=3x
1 – 8x – 5
=3x
Write original equation.
x(x – 5) 1– 8x – 5( )= x(x – 5) 3
xMultiply each side by the LCD, x(x–5).
x(x –5) – 8x = 3(x – 5) Simplify.
x2 – 5x – 8x = 3x – 15 Simplify.
x2 – 16x +15 = 0 Write in standard form.
(x – 1)(x – 15) = 0Factor.
x = 1 or x = 15 Zero product property
EXAMPLE 4 Solve a rational equation with two solutions
The solutions are 1 and 15. Check these in the original equation.
ANSWER
EXAMPLE 5 Check for extraneous solutions
Solve: 6x – 3 =
8x2
x2 – 9– 4x
x + 3
SOLUTION
Write each denominator in factored form. The LCD is (x + 3)(x – 3).
=8x2
(x + 3)(x – 3)–
4xx + 3
6x –3
x + 3(x + 3)(x – 3) 6
x –3= (x + 3)(x – 3) 8x2
(x + 3)(x – 3)(x + 3)(x – 3) 4x
6(x + 3) = 8x2 – 4x(x – 3)
6x + 18 = 8x2 – 4x2 + 12x
EXAMPLE 5 Check for extraneous solutions
0 = 4x2 + 6x –18
0 = 2x2 + 3x – 9
0 = (2x – 3)(x + 3)
2x – 3 = 0 or x + 3 = 0
x = 32
or x = –3
You can use algebra or a graph to check whether either of the two solutions is extraneous.
The solution checks, but the apparent solution –3 is extraneous, because substituting it in the equation results in division by zero, which is undefined.
32
Algebra
GUIDED PRACTICE for Examples 3, 4 and 5
Solve the equation by using the LCD. Check for extraneous solutions.
7
2+
3
x= 35.
SOLUTION
Write each denominator in factored form. The LCD is 2x
7
2+
3
x= 3
2x + 2x = 2x 37
2
3
x
7x + 6 = 6x
x = – 6
GUIDED PRACTICE for Examples 3, 4 and 5
2
x+
4
3= 26.
SOLUTION
Write each denominator in factored form. The LCD is 3x
2
x+
4
3= 2
3x + 3x = 3x 22
x
4
3
6 + 4x = 6x
6 = 2x
x = 3
GUIDED PRACTICE for Examples 3, 4 and 5
3
7+
8
x= 17.
SOLUTION
Write each denominator in factored form. The LCD is 7x
3
7+
8
x= 1
7x + 7x = 7x 13
7
8
x
3x + 56 = 7x
56 = 4x
x = 14
GUIDED PRACTICE for Examples 3, 4 and 5
8. 3
2+ 4
x –1 x –1x +1=
SOLUTION
Write each denominator in factored form. The LCD is 2( x – 1)
3
2+ 4
x –1 x –1x +1=
(x – 1 )(2) + (x – 1)(2) = (x – 1)(2) 32
4x –1
x + 1x 1
3x – 3 + 8= 2x + 2
x = – 3
GUIDED PRACTICE for Examples 3, 4 and 5
3xx +1
– 52x
= 32x
9.
SOLUTION
Write each denominator in factored form. The LCD is (x + 1)(2x)
3x 5 3x +1
–2x
=2x
32 x
2x (x + 1) – 2x (x +1) = 2x (x +1) 3xx +1
52 x
GUIDED PRACTICE for Examples 3, 4 and 5
6x2 – 5x – 5 = 3x + 3
0 = 3x + 3 – 6x2 +5x + 5
0 = – 6x2 + 8x + 8
0 = (3x +2) (x – 2)
x – 2 = 0
x = 2or
3x + 2 = 0
x = 2
3–
GUIDED PRACTICE for Examples 3, 4 and 5
5xx –2
= 7 +10x –2
10.
SOLUTION
Write each denominator in factored form. The LCD is x – 2
5xx –2
= 7 +10x –2
x – 2 = (x – 2) 7 + (x – 2)
10x – 2
5xx –2
5x = 7x – 14 + 10
4 = 2x
x = 2
x=2 results in no solution.