Polynomial equation question?

Question

The roots of the equation 6x^3+ 17x^2- 5x- 6=0 are represented by a,b,c (from least to greatest). Determine an equation whose roots are a+b, a/b, and ab.

The answer is: x^3- 4x^2- 69/4x +63/2=0.

How do I get this?

Answer 1

EASY WAY (graphing calculator):

1. graph the equation

2. find the roots (they are -3 -1/2, 2/3)

3. take the smaller two

4. you want a + b, a/b, and ab. these are -7/2, 6, and 3/2.

5. (x + 7/2)*(x - 6)*(x - 3/2) distribute it out and that's your answer

 

 

HARD WAY (no graphing calculator):

 

ax^3 + bx^2 + cx + d = 0

 

 

possible roots formula is x = ± p/q, where p are factors of d and q are factors of a. 

 

 

1. define formula

6x^3+ 17x^2- 5x- 6 = 0

 

2. define a and d

 

a = 6

d = -6

 

3. find all factors of a and d

p = ±1, ±2, ±3, ±6

q = ±1, ±2, ±3, ±6

 

4. find all possible roots

±p/q = ±1, ±2, ±3, ±6, ±1/2, ±1/3, ±1/6, ±2/3, ±3/2

 

5. try them out

If you try them all out you'll find the roots are -3, -1/2, and 2/3. 

 

 

they want the smaller two: -3, -1/2

 

a + b = -7/2

a/b = 6

ab = 3/2

 

y = (x + 7/2)(x - 6)(x - 3/2) = x^3- 4x^2- 69/4x +63/2