Exercise 135 Page 295

Practice makes perfect

We want to subtract the rational expressions. Notice that both of these fractions have the same denominator. This means they can be subtracted immediately.

x^2/x-5-25/x-5

SubFrac

Subtract fractions

x^2-25/x-5

Now we see that the numerator is showing a difference of squares. This means we can factor it and continue to simplify.

x^2-25/x-5

WritePow

Write as a power

x^2-5^2/x-5

FacDiffSquares

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

(x+5)(x-5)/x-5

ReduceFrac

a/b=.a /(x-5)./.b /(x-5).

x+5

Like in Part A, we have the same denominator in both fraction. Therefore, we can add them immediately.

a^2/a+5+10a+25/a+5

AddFrac

Add fractions

a^2+10a+25/a+5

Notice that the numerator is a perfect square trinomial. Therefore, we can factor it and continue simplifying.

a^2+10a+25/a+5

WritePow

Write as a power

a^2+10a+5^2/a+5

SplitIntoFactors

Split into factors

a^2+2(a)(5)+5^2/a+5

FacPosPerfectSquare

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

(a+5)^2/a+5

ReduceFrac

a/b=.a /(a+5)./.b /(a+5).

a+5

Like in previous parts, we can subtract the rational expressions immediately since they have the same denominator. When we do, we notice that the numerator forms a perfect square trinomial.

x^2/x-y-2xy-y^2/x-y

SubFrac

Subtract fractions

x^2-(2xy-y^2 )/x-y

Distr

Distribute -1

x^2-2xy+y^2/x-y

FacNegPerfectSquare

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

(x-y)^2/x-y

ReduceFrac

a/b=.a /(x-y)./.b /(x-y).