Polynomials

Question

Given the polynomial p(x)=2x³-5x²-6x+4

a. Factor the polynomial completely into linear factors

b. List all the zeros of p(x) (in exact form)

 

Please explain how you got the answers

Thanks!

Answer 1

To solve this type of problem, its usually good to use Descartes' rule of signs to determine the possible combination of roots.  For this example, f(x) has 2 changes of sign, indicating that there are either two positive real roots or no positive real roots.  f(-x) has 1 change of sign, indicating that there is one negative real root.

Now, use the rational roots theorem to find all of the possible rational roots (+/- factors of the constant term divided by factors of the leading coefficient).  The factors in this case are +/- 1,+/- 2, +/- 4, and +/- 1/2.

Now find the values for f(x) for each of these values.

f(1) = -5

f(2) = -12

f(4) = 28

f(-1) = 3

f(-2) = -20

no need to determine f(-4) because bounds for all three roots have been found.  One is between 1 and 2; one is between 2 and 4; and one is between -1 and -2.  These conclusions are made using the intermediate value theorem.

Now, the only possible rational root still worth testing is 1/2:

f(1/2) = 0.  Finally!

 

Now use synthetic division to divide out this root, leaving 2x²-4x-8.  This can be factored using the quadratic formula.  

The polynomial is factored as (2x - 1)(x - 1 - √5)(x -1 + √5).