Exercise 78 Page &nbsp; 145

Factor out x

x(x-4)-4x+16

Factor out 4

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

Factor out (x-4)

(x-4)(x-4)

Let's continue with the denominator.

3x^2-10x-8

WriteDiff

Write as a difference

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

Factor out x

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

Factor out 4

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

Factor out (3x+2)

(x-4)(3x+2)

Now we can rewrite the fraction as follows. (x-4)(x-4)/(x-4)(3x+2) Note that (x-4) is in both the numerator and denominator.

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

Write as a product of fractions

x-4/x-4*x-4/3x+2

a/a=1

1*x-4/3x+2

Identity Property of Multiplication

x-4/3x+2

The 1 created when dividing (x-4) by (x-4) is sometimes referred to as a Giant One. The fraction can be rewritten as follows. x-4/3x+2

b Let's factor both the numerator and denominator. The numerator is 10x+25, which means that 5 can be factored out.

10x+25 = 5(2x+5) For the denominator, let's rewrite - x as -6x+5x and then factor.

2x^2-x-15

WriteSum

Write as a sum

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

Factor out 2x

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

Factor out 5

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

Factor out (x-3)

(2x+5)(x-3)

Now we have both numerator and denominator in factored form, which means we can rewrite the fraction as follows. 10x+25/2x^2-x-15=5(2x+5)/(2x+5)(x-3) Here, (2x+5) is both above and below the fraction line, meaning they can be used to create a Giant One.

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

CommutativePropMult

Commutative Property of Multiplication

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

Write as a product of fractions

5/x-3* 2x+5/2x+5

a/a=1

5/x-3* 1

Identity Property of Multiplication

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.

(k-4)(2k+1)/5(2k+1)

Write as a product of fractions

k-4/5* 2k+1/2k+1

a/a=1

k-4/5*1

Identity Property of Multiplication

k-4/5

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

(k-3)(k-4)/10(k-3)

CommutativePropMult

Commutative Property of Multiplication

(k-4)(k-3)/10(k-3)

Write as a product of fractions

k-4/10* k-3/k-3

a/a=1

k-4/10* 1

Identity Property of Multiplication

k-4/10

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.

(k-4)(2k+1)/5(2k+1)÷(k-3)(k-4)/10(k-3)

SubstituteExpressions

Substitute expressions

k-4/5÷k-4/10

DivFracByFracDiv

a/b÷c/d=a/b*d/c

k-4/5*10/k-4

CommutativePropMult

Commutative Property of Multiplication

k-4/k-4*10/5