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Core Connections Integrated III, 2015 View details

2. Section 11.2

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Exercise 70 Page 586

a The fractions we want to add do not have the same denominator.

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

Let's factor

x^{2} - 4

to see if we can find a common denominator.

x^{2} - 4

WritePow

Write as a power

x^{2} - 2^{2}

FacDiffSquares

$a^{2} - b^{2} = (a + b) (a - b)$

(x + 2) (x - 2)

This means that we can rewrite the expression as follows.

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

By multiplying the numerator and denominator of the first fraction by

(x - 2),

we will be able to add the fractions and then simplify.

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

ExpandFrac

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

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

AddFrac

Add fractions

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

AddTerms

Add terms

\frac{x + 1}{( x + 2 ) ( x - 2 )}

b To find how to rewrite the fractions so that they have the same denominator, let's factor the denominators. In the first fraction we can factor out

2 .

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

In the second fraction let's rewrite

4 x

2 x + 2 x

and factor.

x^{2} + 4 x + 4

▼

Factor

WriteSum

Write as a sum

x^{2} + 2 x + 2 x + 4

FactorOut

Factor out $x$

x (x + 2) + 2 x + 4

FactorOut

Factor out $2$

x (x + 2) + 2 (x + 2)

FactorOut

Factor out $(x + 2)$

(x + 2) (x + 2)

Therefore, the expression can be written as follows.

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

By multiplying the first numerator and denominator by

(x + 2)

and the second numerator and denominator by

2,

the fractions will have the same denominator. Then we can subtract.

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

ExpandFrac

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

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

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 2}{b \cdot 2}$

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

SubFrac

Subtract fractions

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

Distr

Distribute $3$

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

SubTerm

Subtract term

\frac{x + 6}{2 ( x + 2 ) ( x + 2 )}

ProdToPowTwoFac

$a \cdot a = a^{2}$

\frac{x + 6}{2 ( x + 2 ) ^{2}}

c We now have two fractions that are being multiplied.

\frac{x ^{2} + 5 x + 6}{x ^{2} - 9} \cdot \frac{x - 3}{x ^{2} + 2 x}

Before we multiply, let's factor the polynomials.

x^{2} + 5 x + 6

▼

Factor

WriteSum

Write as a sum

x^{2} + 2 x + 3 x + 6

FactorOut

Factor out $x$

x (x + 2) + 3 x + 6

FactorOut

Factor out $3$

x (x + 2) + 3 (x + 2)

FactorOut

Factor out $(x + 2)$

(x + 3) (x + 2)

The first denominator can be factored using the fact that it's a difference of squares.

x^{2} - 9

WritePow

Write as a power

x^{2} - 3^{2}

FacDiffSquares

$a^{2} - b^{2} = (a + b) (a - b)$

(x + 3) (x - 3)

In the second denominator we factor out an

x .

x^{2} + 2 x = x (x + 2)

Now we substitute the factor forms into the original expression and multiply.

\frac{x ^{2} + 5 x + 6}{x ^{2} - 9} \cdot \frac{x - 3}{x ^{2} + 2 x}

SubstituteExpressions

Substitute expressions

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

MultFrac

Multiply fractions

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

CommutativePropMult

Commutative Property of Multiplication

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

WriteProdFrac

Write as a product of fractions

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

QuotOne

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

\frac{1}{x} \cdot 1

IdPropMult

Identity Property of Multiplication

\frac{1}{x}

The expression can be simplified to

\frac{1}{x} .

d When dividing two fractions the denominator is inverted and multiplied by the numerator.

\frac{4}{x - 2} \div \frac{8}{2 - x}

DivFracByFracDiv

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$

\frac{4}{x - 2} \cdot \frac{2 - x}{8}

MultFrac

Multiply fractions

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

FactorOutNegSignSwap

$a - b = - (b - a)$

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

▼

Simplify

CommutativePropMult

Commutative Property of Multiplication

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

WriteProdFrac

Write as a product of fractions

\frac{4}{- 8} \cdot \frac{2 - x}{2 - x}

QuotOne

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

\frac{4}{- 8} \cdot 1

IdPropMult

Identity Property of Multiplication

\frac{4}{- 8}

MoveNegDenomToFrac

Put minus sign in front of fraction

- \frac{4}{8}

CalcQuot

Calculate quotient

- 0.5

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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