Exercise 35 Page 688

The result of adding or subtracting rational expressions is also a rational expression. A rational expression is in simplest form when its numerator and denominator have no common factors.

Rational Expression	Factors of Numerator	Factors of Denominator	Simplest Form?
4/x^2	2* 2	x* x	Yes ✓
2x/x^2	2* x	x* x	No *
x+1/x^2-1	x+1	( x+1)*(x-1)	No *

We are wondering whether using the least common denominator (LCD) to add or subtract rational expressions ensures that the answer is in simplest form. Let's consider the following example. 2x^2+5x+2/x^2+x-1/x+1Since the rational expressions have different denominators, we can use the LCD to rewrite them. Denominator1:& x^2+x= x( x+1) Denominator2:& x+1 LCD:& x( x+1)=x^2+x We have that the LCD is the same as the first denominator, x^2+x. Let's rewrite the second rational expression so that we can find the result of the subtraction.

2x^2+5x+2/x^2+x-1/x+1

ExpandFrac

a/b=a * x/b * x

2x^2+5x+2/x^2+x-1* x/(x+1) x

IdPropMult

Identity Property of Multiplication

2x^2+5x+2/x^2+x-x/(x+1)x

Distr

Distribute x

2x^2+5x+2/x^2+x-x/x^2+x

SubFrac

Subtract fractions

2x^2+5x+2-x/x^2+x

SubTerm

Subtract term

2x^2+4x+2/x^2+x

Now we will factor the numerator and the denominator to see if they have any common factors. If they do, it means that our answer is not in simplest form.

2x^2+4x+2/x^2+x

▼

Factor the Numerator

FactorOut

Factor out 2

2(x^2+2x+1)/x^2+x

FacPosPerfectSquare

a^2+2ab+b^2=(a+b)^2

2(x+1)^2/x^2+x

PowToProdTwoFac

a^2=a* a

2(x+1)(x+1)/x^2+x

FactorOut

Factor out x

2(x+1)(x+1)/x(x+1)

We can see that the numerator and the denominator have one common factor. Canceling out the common factor will give us the answer in simplest form.

2(x+1)(x+1)/x(x+1)

CancelCommonFac

Cancel out common factors

2(x+1)(x+1)/x(x+1)

SimpQuot

Simplify quotient

2(x+1)/x

Distr

Distribute 2

2x+2/x

This example shows us that using the LCD to add or subtract rational expressions does not mean that the answer will be in simplest form. It is because the numerator of the result may contain a factor of the LCD.