Exercise 58 Page 582

a When multiplying fractions, the numerators are multiplied with each other, and the denominators are multiplied with each other separately. The expression can then by simplified by finding Giant Ones.

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

MultFrac

Multiply fractions

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

PowToProdTwoFac

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

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

CommutativePropMult

Commutative Property of Multiplication

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

WriteProdFrac

Write as a product of fractions

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

QuotOne

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

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

IdPropMult

Identity Property of Multiplication

\frac{x - 3}{3 x - 1 4}

b When dividing two fractions, the first fraction is multiplied with the second fraction's reciprocal.

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

DivFracByFracDiv

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

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

MultFrac

Multiply fractions

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

To simplify the fraction, we need to factor the trinominals. Let's start with the first parentheses in the numerator.

4 x^{2} + 5 x - 6

▼

Factor

WriteSum

Write as a sum

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)

It was possible to write the expression as two factors. We will now do the same thing with the next parentheses.

6 x^{2} - 5 x + 1

▼

Factor

WriteDiff

Write as a difference

6 x^{2} - 3 x - 2 x + 1

FactorOut

Factor out $3 x$

3 x (2 x - 1) - 2 x + 1

FactorOut

Factor out $- 1$

3 x (2 x - 1) - 1 (2 x - 1)

FactorOut

Factor out $(2 x - 1)$

(3 x - 1) (2 x - 1)

Now we move down to the denominator for some more factoring!

3 x^{2} + 5 x - 2

▼

Factor

WriteSum

Write as a sum

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

FactorOut

Factor out $3 x$

3 x (x + 2) - x - 2

FactorOut

Factor out $- 1$

3 x (x + 2) - 1 (x - 2)

FactorOut

Factor out $(x + 2)$

(3 x - 1) (x + 2)

Let's factor the final one as well.

4 x^{2} + x - 3

▼

Factor

WriteSum

Write as a sum

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

FactorOut

Factor out $4 x$

4 x (x + 1) - 3 x - 3

FactorOut

Factor out $- 3$

4 x (x + 1) - 3 (x + 1)

FactorOut

Factor out $(x + 1)$

(4 x - 3) (x + 1)

We can now use the factored form of each parentheses to simplify the fractions.

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

SubstituteExpressions

Substitute expressions

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

CommutativePropMult

Commutative Property of Multiplication

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

WriteProdFrac

Write as a product of fractions

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

QuotOne

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

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

IdPropMult

Identity Property of Multiplication

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

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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