Exercise 7 Page 565

Practice makes perfect

a To reduce a fraction we have to rewrite the numerator and denominator to identify common factors. To simplify the expression we want to factor both the numerator and denominator. We will start with the numerator.

x^{2} - 8 x + 16

WriteDiff

Write as a difference

x^{2} - 4 x - 4 x + 16

FactorOut

Factor out $x$

x (x - 4) - 4 x + 16

FactorOut

Factor out $4$

x (x - 4) - 4 (x - 4)

FactorOut

Factor out $(x - 4)$

(x - 4) (x - 4)

Let's continue with the denominator.

3 x^{2} - 10 x - 8

WriteDiff

Write as a difference

3 x^{2} + 2 x - 12 x - 8

FactorOut

Factor out $x$

x (3 x + 2) - 12 x - 8

FactorOut

Factor out $4$

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

FactorOut

Factor out $(3 x + 2)$

(x - 4) (3 x + 2)

Now we can rewrite the fraction as follows.

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

Note that

(x - 4)

is in both the numerator and denominator.

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

WriteProdFrac

Write as a product of fractions

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

QuotOne

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

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

IdPropMult

Identity Property of Multiplication

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

The 1 created when dividing

(x - 4)

(x - 4)

is sometimes referred to as a Giant One. The fraction can be rewritten as follows.

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

b Let's factor both the numerator and denominator. The numerator is

10 x + 25,

which means that

5

can be factored out.

10 x + 25 = 5 (2 x + 5)

For the denominator, let's rewrite

- x

- 6 x + 5 x

and then factor.

2 x^{2} - x - 15

WriteSum

Write as a sum

2 x^{2} - 6 x + 5 x - 15

FactorOut

Factor out $2 x$

2 x (x - 3) + 5 x - 15

FactorOut

Factor out $5$

2 x (x - 3) + 5 (x - 3)

FactorOut

Factor out $(x - 3)$

(2 x + 5) (x - 3)

Now we have both numerator and denominator in factored form, which means we can rewrite the fraction as follows.

\frac{1 0 x + 2 5}{2 x ^{2} - x - 1 5} = \frac{5 ( 2 x + 5 )}{( 2 x + 5 ) ( x - 3 )}

Here,

(2 x + 5)

is both above and below the fraction line, meaning they can be used to create a Giant One.

\frac{5 ( 2 x + 5 )}{( 2 x + 5 ) ( x - 3 )}

CommutativePropMult

Commutative Property of Multiplication

\frac{5 ( 2 x + 5 )}{( x - 3 ) ( 2 x + 5 )}

WriteProdFrac

Write as a product of fractions

\frac{5}{x - 3} \cdot \frac{2 x + 5}{2 x + 5}

QuotOne

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

\frac{5}{x - 3} \cdot 1

IdPropMult

Identity Property of Multiplication

\frac{5}{x - 3}

c Let's first look at the fraction in the numerator and denominator separately, then simplify them. We start with the numerator.

\frac{( k - 4 ) ( 2 k + 1 )}{5 ( 2 k + 1 )}

WriteProdFrac

Write as a product of fractions

\frac{k - 4}{5} \cdot \frac{2 k + 1}{2 k + 1}

QuotOne

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

\frac{k - 4}{5} \cdot 1

IdPropMult

Identity Property of Multiplication

\frac{k - 4}{5}

Now, we move on to the fraction in the denominator.

\frac{( k - 3 ) ( k - 4 )}{1 0 ( k - 3 )}

CommutativePropMult

Commutative Property of Multiplication

\frac{( k - 4 ) ( k - 3 )}{1 0 ( k - 3 )}

WriteProdFrac

Write as a product of fractions

\frac{k - 4}{1 0} \cdot \frac{k - 3}{k - 3}

QuotOne

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

\frac{k - 4}{1 0} \cdot 1

IdPropMult

Identity Property of Multiplication

\frac{k - 4}{1 0}

Now that we have simplified the expressions, let's perform the division. When a fraction is divided by another fraction. The division sign becomes a multiplication sign and the denominator fraction is inverted, meaning the numerator and denominator switch places.