Use synthetic division to show that the given value of x is a solution of the third-degree polynomial eq, and use the result to factor the polynomial completely.

Question

List all real solutions of the equation. (Enter your answers as a comma-separated list.)

48x³ - 224x² + 173x - 30 = 0 , x = 2/3

x =

Answer 1

2/3 | 48 -224 173 -30

32 -128 30

48 -192 45 0

Use the Quadratic Formula to determine the other two roots.

Of you could use the Rational Root Theorem.

Answer 2

p = +-{1,2,3,5,6,10,15,30}

q = +-{1,2,3,4,6,8,12,16,24,48}

1st it is VERY important to note that

Per rational root theorem the POTENTIAL,

POSSIBLE rational solutions are:

p/q = +-{1,2,3,5,6,10,12,15,30,

1/2, 1/3, 1/4, 1/6, 1/8, 1/12, 1/16,1/24, 1/48,

2/3,3/2,3/4, 3/8, 3/16,4/3,10/3,5/2,5/3,5/8,

5/12,5/16,5/24,5/48,15/16,...}

They are GRACIOUS, GENEROUS, and KIND enough to tell us that

2/3 is a solution, which appears in the list! Otherwise

will would have to test ALL of them! Saves us ALOT of time,

effort, and energy! THank you

Since x=2/3 is a solution then 3x=2 ---> 3x-2=0 is a factor

Synthetic division

2/3 | 48 -224 173 -30

32 -128 30

_____________________________

48 -192 45 0

The quadratic quotient is 48x^2 - 192x + 45 = 0

divides by 3: 16x^2 - 64x + 15 = 0

( 4x - 15)(4x - 1 ) = 0

4x-15=0 or 4x-1= 0

4x=15 or 4x=1

x = 15/4 or x = 1/4

The three solutions are { 2/3, 15/4, 1/4} all of which

are in the list