Chapter Closure
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Exercise 131 Page 251
a Create a common denominator, then add the rational expressions. Can you find any common factors?
b Create a common denominator, then subtract the rational expressions. Can you find any common factors?
c Find common factors in the numerator and the denominator.
d Create a common denominator, then add the rational expressions. Can you find any common factors?
a See solution.
b 1/x-1, x≠ 1
c 4(x+3)/x-4, x≠ - 2, 0, or 4
d 1/4x, x≠ 2, 0
a Before we add the fractions, let us think about forbidden values of x. Since we cannot divide by 0, the expressions in the denominators cannot be 0. Notice that the square of a number is always nonnegative. Therefore, x^2+9 must always be positive.
5x/x+3 + 3+x/x^2+9
ExpandFrac
a/b=a * ( x^2+9 )/b * ( x^2+9 )
5x( x^2+9 )/(x+3)( x^2+9 ) + 3+x/x^2+9
ExpandFrac
a/b=a * (x+3)/b * (x+3)
5x( x^2+9 )/(x+3)( x^2+9 ) + (3+x)(x+3)/( x^2+9 ) (x+3)
AddFrac
Add fractions
5x( x^2+9 )+(3+x)(x+3)/(x+3)( x^2+9 )
▼
Simplify numerator
Distr
Distribute 5x
5x^3+45x+(3+x)(x+3)/(x+3)( x^2+9 )
CommutativePropAdd
Commutative Property of Addition
5x^3+45x+(x+3)(x+3)/(x+3)( x^2+9 )
ProdToPowTwoFac
a* a=a^2
5x^3+45x+(x+3)^2/(x+3)( x^2+9 )
ExpandPosPerfectSquare
(a+b)^2=a^2+2ab+b^2
5x^3+45x+x^2+6x+9/(x+3)( x^2+9 )
AddTerms
Add terms
5x^3+51x+x^2+9/(x+3)( x^2+9 )
CommutativePropAdd
Commutative Property of Addition
5x^3+x^2+51x+9/(x+3)( x^2+9 )
b Similarly as in Part A, we know that the denominator cannot be 0. Therefore we know that x≠ 1. To subtract the 1, we have to rewrite it as a fraction with a common denominator to the given fraction.
c We want to multiply the given rational expressions.
x^2-4x=0
▼
Solve for x
SplitIntoFactors
Split into factors
x* x-4* x=0
FactorOut
Factor out x
x(x-4)=0
ZeroProdProp
Use the Zero Product Property
lx=0 x-4=0
AddEqn
(II): LHS+4=RHS+4
lx_1=0 x_2=4
x^2+5x+6/x^2-4x* 4x/x+2
MultFrac
Multiply fractions
(x^2+5x+6)4x/(x^2-4x)(x+2)
SplitIntoFactors
Split into factors
(x^2+5x+6)4x/(x* x-4* x)(x+2)
FactorOut
Factor out x
(x^2+5x+6)4x/x(x-4)(x+2)
x^2+5x+6
SplitIntoFactors
Split into factors
x^2+5x+(2)(3)
WriteSum
Write as a sum
x^2+( 2+ 3)x+( 2)( 3)
Write as (x+ a)(x+ b)
(x+ 2)(x+ 3)
(x+2)(x+3)4x/x(x-4)(x+2)
ReduceFrac
a/b=.a /(x+2)./.b /(x+2).
(x+3)4x/x(x-4)
ReduceFrac
a/b=.a /x./.b /x.
(x+3)4/x-4
Distr
Distribute 4
4x+12/x-4
d We want to divide the given rational expressions.
x^2-4x+4=0
▼
Factor
WritePow
Write as a power
x^2-4x+2^2=0
SplitIntoFactors
Split into factors
x^2-2(x)(2)+2^2=0
FacNegPerfectSquare
a^2-2ab+b^2=(a-b)^2
(x-2)^2=0
x=2
x^2-2x/(x-2)^2÷ 4x^2/x-2
DivFracByFracDiv
a/b÷c/d=a/b*d/c
x^2-2x/(x-2)^2* x-2/4x^2
MultFrac
Multiply fractions
(x^2-2x)(x-2)/(x-2)^2 4x^2
▼
Factor
x(x-2)(x-2)/(x-2)^2 4x^2
ProdToPowTwoFac
a* a=a^2
x(x-2)^2/(x-2)^2 4x^2
ReduceFrac
a/b=.a /x./.b /x.
(x-2)^2/(x-2)^2 4x
ReduceFrac
a/b=.a /(x-2)^2./.b /(x-2)^2.
1/4x
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