Exercise 57 Page 582

a To add the rational expressions they need to have the same denominator. It's easier to see what we need to multiply the fractions with if we factor the denominator in the first fraction.

\frac{4 x}{x ^{2} - 2 x - 8} + \frac{4}{x - 4}

Let's factor it.

x^{2} - 2 x - 8

▼

Factor

WriteSum

Write as a sum

x^{2} - 4 x + 2 x - 8

FactorOut

Factor out $x$

x (x - 4) + 2 x - 8

FactorOut

Factor out $2$

x (x - 4) + 2 (x - 4)

FactorOut

Factor out $(x - 4)$

(x + 2) (x - 4)

The sum of the fractions can be written as follows.

\frac{4 x}{x ^{2} - 2 x - 8} + \frac{4}{x - 4}

To get the same denominator in the second fraction we can multiply by

(x + 2) .

But, in order to not change the expression we need to multiply by

(x + 2)

in both the numerator and the denominator.

\frac{4 x}{( x + 2 ) ( x - 4 )} + \frac{4}{x - 4}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot ( x + 2 )}{b \cdot ( x + 2 )}$

\frac{4 x}{( x + 2 ) ( x - 4 )} + \frac{4 ( x + 2 )}{( x - 4 ) ( x + 2 )}

AddFrac

Add fractions

\frac{4 x + 4 ( x + 2 )}{( x + 2 ) ( x - 4 )}

Distr

Distribute $4$

\frac{4 x + 4 x + 8}{( x + 2 ) ( x - 4 )}

AddTerms

Add terms

\frac{8 x + 8}{( x + 2 ) ( x - 4 )}

b The fractions do not have the same denominator, so similar to the previous exercise we will factor

4 x^{2} + 5 x - 6 .

\frac{1 6 x - 1 2}{4 x ^{2} + 5 x - 6} - \frac{3}{x + 2}

Let's factor it.

4 x^{2} + 5 x - 6

▼

Factor

WriteDiff

Write as a difference

4 x^{2} + 8 x - 3 x - 6

FactorOut

Factor out $4 x$

4 x (x + 2) - 3 x - 6

FactorOut

Factor out $3$

4 x (x + 2) - 3 (x + 2)

FactorOut

Factor out $(x + 2)$

(4 x - 3) (x + 2)

This means that we can rewrite the expression as follows.

\frac{1 6 x - 1 2}{( 4 x - 3 ) ( x + 2 )} - \frac{3}{x + 2}

By multiplying both numerator and denominator of the second fraction by

(4 x - 3),

both fractions get the same denominator.

\frac{1 6 x - 1 2}{( 4 x - 3 ) ( x + 2 )} - \frac{3}{x + 2}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot ( 4 x - 3 )}{b \cdot ( 4 x - 3 )}$

\frac{1 6 x - 1 2}{( 4 x - 3 ) ( x + 2 )} - \frac{3 ( 4 x - 3 )}{( x + 2 )}

SubFrac

Subtract fractions

\frac{1 6 x - 1 2 - 3 ( 4 x - 3 )}{( 4 x - 3 ) ( x + 2 )}

Distr

Distribute $- 3$

\frac{1 6 x - 1 2 - 1 2 x + 9}{( 4 x - 3 ) ( x + 2 )}

AddSubTerms

Add and subtract terms

\frac{4 x - 3}{( 4 x - 3 ) ( x + 2 )}

WriteProdFrac

Write as a product of fractions

\frac{4 x - 3}{4 x - 3} \cdot \frac{1}{x + 2}

QuotOne

$\frac{a}{a} = 1$

1 \cdot \frac{1}{x + 2}

IdPropMult

Identity Property of Multiplication

\frac{1}{x + 2}

Subchapter links

11.2.1 Creating a Three-Dimensional Model p.582-583

11.2.2 Graphing Equations in Three Dimensions p.586-587

11.2.3 Solving Systems of Three Equations with Three Variables p.590-592

11.2.4 Using Systems of Three Equations for Curve Fitting p.598-600

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